# Classification of general sequences by frame-related operators.

1 Introduction

Frames [12, 6, 10] have become an important topic in applied mathematics in the last decades. They are also used in many engineering applications, for example in signal processing [5]. Clearly, they have some important advantages compared to orthonormal bases. However, even the frame condition cannot always be satisfied for the whole space, and so other classes of sequences have been investigated, for example, frame sequences, Bessel sequences, lower frame sequences, and Riesz-Fischer sequences [4, 7, 9, 10]. For such sequences, which need not be frames in general, the frame-related operators, i.e. the analysis, the synthesis and the frame operator can still be defined, see e.g. [7, 9]. In general, these operators can be unbounded.

In this note we want to give an overview of the connection between the properties of those operators and those of the sequences. We collect some existing results [7, 10, 18, 19], generalize results for particular systems, like frames of translates [23, 24], Gabor [26] or wavelet frames [25], to the general frame setting, and add new, original results. Some known results are proved here for the sake of completeness, in particular, when there is no proof in the literature. Note that one result was obtained independently in [15]. While the paper [15] focuses on the investigation of sufficient conditions for the closability of the synthesis operator, the main aim of the present paper is to give a general overview of all the associated operators.

Our notation and some preliminary results are given in Section 2. In Section 3 we consider the operators associated to an arbitrary sequence, namely, the analysis, synthesis and 'frame' operators, as well as the operator based on the Gram matrix, and we investigate their properties. Section 4 is devoted to the characterization of sequences (Bessel and frame sequences, frames, Riesz bases, lower frame sequences, Riesz-Fischer sequences, complete sequences) via the associated operators. We consider operators that preserve the sequence type and finally, we classify sequence types via orthonormal bases.

2 Notation and Preliminaries

Throughout the paper we consider a sequence [PSI] = [([psi].sub.k]).sup.[infinity].sub.k=1] with elements from a (infinite dimensional) Hilbert space (H, <.,.>). The notation span {[PSI]} is used to denote the linear span of [([[psi].sub.k]).sup.[infinity].sub.k=1] and [bar.span] {[PSI]} denotes the closed linear span of [([[psi].sub.k]).sup.[infinity].sub.k=1]. The sequence [([e.sub.k]).sup.[infinity].sub.k=1] denotes an orthonormal basis of H and [([[delta].sub.k]).sup.[infinity].sub.k=1] denotes the canonical basis of [l.sup.2]. The notion operator is used for a linear mapping. Given an operator F, we denote its domain by dom(F), its range by ran(F) and its kernel by ker(F). First we recall the definitions of the basic concepts used in the paper.

Definition 2.1 The sequence [PSI] is called

* complete in H if [bar.span] {[PSI]} = H;

* a Bessel sequence for H with bound B if 0 < B < [infinity] and [[summation].sup.[infinity]].sub.k=1] [[absolute value of <f, [[psi].sub.k]>].sup.2] [less than or equal to] B [[parallel]f[parallel].sup.2] for every f [member of] H;

* a lower frame sequence for H with bound A if A > 0 and A [[parallel]f[parallel].sup.2] [less than or equal to] [[summation].sup.[infinity]].sub.k=1] [[absolute value of <f, [[psi].sub.k]>.sup.2] for every f [member of] H;

* a frame for H with bounds A, B if it is a Bessel sequence for H with bound B and a lower flame sequence for H with bound A;

* a frame-sequence if it is a frame for its closed linear span;

* a Riesz basis for H with bounds A, B if [PSI] is complete in H, A > 0, B < [infinity], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all finite scalar sequences ([c.sub.k]) (and hence, for all [([c.sub.k]).sup.[infinity]].sub.k=1] [member of] [l.sup.2]);

* a Riesz-Fischer sequence with bound A if A > 0 and A [summation] [[absolute value of [c.sub.k]].sup.2] [less than or equal to] [[parallel][summation][c.sub.k][[psi].sub.k][parallel].sup.2] for all finite scalar sequences ([c.sub.k]) (and hence, for all [([c.sub.k]).sup.[infinity]].sub.k=1] [member of] [l.sup.2] such that [[summation].sup.[infinity]].sub.k = 1] [c.sub.k][[psi].sub.k] converges in H).

Lemma 2.2 Let [([d.sub.k]).sup.[infinity]].sub.k=1] be a sequence such that

[[infinity].summation over [k=1]][c.sub.k][d.sub.k] converges for all [([c.sub.k]).sup.[infinity]].sub.k=1] [member of] [l.sup.2]. (1)

Then [([d.sub.k]).sup.[infinity]].sub.k=1] [member of] [l.sup.2].

Proof: If (1) is assumed to hold with absolute convergence of the series, the above statement is proved in [17, [section]30 1.(5)]. Similar proof (using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] leads to the validity of the above lemma.

We also need the following well-known results:

Proposition 2.3 Let an operator F be densely defined. Then

(a) [11, Prop. X.1.6] F* is closed; in particular, every self-adjoint operator is closed;

(b) [11, Prop. X.1.6] F is closable if and only if F* is densely defined;

(c) [22, Th. VIII.1] If F is closable, then its closure is [bar.F] = (F*)* and [bar.F]* = F* ;

(d) [11, Prop. X.1.13] [(ran(F)).sup.[perpendicular]] = ker(F*);

(e) [11, Prop. X.4.2] If F is closed, then F*F is self-adjoint.

3 Associated Operators for Arbitrary Sequences

For any sequence [PSI], the associated analysis operator C, synthesis operator D, 'frame' operator S, and Gram operator G are (possibly unbounded) operators, defined as follows:

C: dom(C) [right arrow] [l.sup.2], Cf = [(<f, [[psi].sub.k]>).sup.[infinity].sub.k=1], where dom(C) = {f [member of] H : [(<f, [[psi].sub.k]>).sup.[infinity].sub.k=1] [member of] [l.sup.2]};

D: dom(D) [right arrow] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where dom(D) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges in H};

S: dom(S) [right arrow] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where dom(S) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges in H};

G: dom(G) [right arrow] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where dom(G) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the Gram matrix [([G.sub.k,l]).sub.kl] is defined by [G.sub.k,l] = <[[psi].sub.l], [[psi].sub.k]>, k, l [member of] N.

3.1 Domains

Lemma 3.1 Given an arbitrary sequence [PSI], the following statements hold.

(i.1) dom(S) [subset or equal to] dom (C).

(i.2) dom(S) = dom(C) if and only if ran(C) [subset or equal to] dom(D).

(ii) dom(D) [subset or equal to] dom(G) if and only if ran(D) [subset or equal to] dom(C).

(iii) D is densely defined.

(iv) [9, Prop. 4.5] If [[summation].sup.[[infinity]].sub.l=1] [[absolute value of [<[[psi].sub.l], [[psi].sub.k]}.sup.2] < [infinity] for every k [member of] N, then C is densely defined.

(v) If [[summation].sup.[[infinity]].sub.l=1] [[absolute value of [<[[psi].sub.l], [[psi].sub.k]>.sup.2] < [infinity] for every k [member of] N, then G is densely defined.

(vi) [[summation].sup.[[infinity]].sub.l=1] [[summation].sup.[[infinity]].sub.k=1] [[absolute value of [<[[psi].sub.l], [[psi].sub.k]}.sup.2] < [infinity], then dom(G) = dom(D) = [l.sup.2], G is a Hilbert-Schmidt operator and [PSI] is a Bessel sequence for H.

(vii) If [[psi].sub.k] [member of] dom(S) for every k [member of] N, then S is densely defined.

Proof: (i.1) Assume that f [member of] dom(S). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges as N [right arrow] [infinity] and thus f [member of] dom(C).

(i.2) If ran(C) [subset or equal to] dom(D), it is obvious that dom(C) [subset or equal to] dom(S) and the other inclusion is given in (i.1). Now assume that dom (S) = dom (C) and take C f [member of] ran(C). Then f [member of] dom(S), which implies that C f [member of] dom(D).

(ii) First observe that if [[summation].sup.[infinity]].sub.l=1] [c.sub.l][[psi].sub.l] converges in H, then [[summation].sup.[infinity]].sub.l=1] [c.sub.l] <[[psi].sub.l], [[psi].sub.k]> converges for every k [member of] N and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that dom(D) [subset or equal to] dom(G) and take Dc [member of] ran(D). Then c [member of] dom(G), which implies that [(<Dc, [[psi].sub.k]>).sup.[infinity].sub.k=1] [member of] [l.sup.2] and thus, Dc [member of] dom(C).

Now assume that ran(D) [subset or equal to] dom(C) and take c [member of] dom(D). Then Dc [member of] dom(C) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, c [member of] dom(G).

(iii) is clear, since the finite sequences are dense in [l.sup.2].

(iv) Since the statement is given in [9] without proof, for the sake of completeness we include a proof here. First observe that [([bar.span]{[PSI]}.sup.[perpendicular]][subset or equal to] dom(C). Assume that [[summation].sup.[infinity]].sub.l=1] [[absolute value of <[[psi].sub.l],[[psi].sub.k].sup.2]] < [infinity] for every k [member of] N. Then span {[PSI]} [subset or equal to] dom(C). Therefore, H = [bar.span]{[PSI]} [direct sum] [([bar.span{[PSI]}).sup.[perpendicular]] [subset or equal to] [bar.dom(C)], which implies that dom(C) is dense in H.

(v) Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every k [member of] N. Then all the canonical vectors [[delta].sub.k] (and thus, all the finite sequences) belong to dom(G). Then the conclusion follows as in (iii).

(vi) Assume that [[summation].sup.[infinity]].sub.l=1] [[absolute value of <[[psi].sub.l], [[psi].sub.k]>].sup.2] < [infinity]. Then G is a Hilbert-Schmidt operator [21, Cor. 16.11]. Furthermore, [PSI] is a Bessel sequence [10, Lemma 3.5.1] (see Prop. 4.4(a2)), which implies that dom(D) = [l.sup.2] [10, Theor. 3.2.3] (see Prop. 4.2).

(vii) Follows in a similar way as in (iv), having in mind that [([bar.span {[PSI]}).sup.[pependicular]] [subset or equal to] dom(S).

Remarks 3.2

(1) Concerning Lemma 3.1(i):

If [PSI] is a Bessel sequence for H, then dom(S) = dom(C) = H. Note that the equality dom(S) = dom(C) might hold even in cases when [PSI] is not a Bessel sequence. Consider for example the non-Bessel sequence

[PSI] = ([e.sub.1], [e.sub.1], [e.sub.2], [e.sub.1], [e.sub.2], [e.sub.3], [e.sub.1], [e.sub.2], [e.sub.3], [e.sub.4],...).

Then dom(C) = {0} and thus, Lemma 3.1(i.1) implies that dom(S) = {0}.

Note that the equality dom(S) = dom(C) might fail. Consider for example the non-Bessel sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As an example of an element which belongs to dom(C) and does not belong to dom(S) consider h [member of] H such that <h, [e.sub.n]> = [2.sup.-2(n-1)], [for all]n [member of] N.

(2) Concerning Lemma 3.1(ii):

If [PSI] is a Bessel sequence for H, then dom(D) = dom(G) = [l.sup.2]. For a non-Bessel sequence [PSI], the inclusion dom(D) [subset or equal to] dom(G) might fail. Take for example [PSI] = ([e.sub.1], [e.sub.2], [e.sub.1], [e.sub.3], [e.sub.1], [e.sub.4],...). Then [[delta].sub.1] [member of] dom(D) and [[delta].sub.1] [not member of] dom(G).

Note that a necessary condition for the validity of dom(G) = dom(D) is that all the canonical vectors are in dom(G), which holds if and only if [[summation].sup.[infinity]].sub.l=1] [[absolute value of <[[psi].sub.l], [[psi].sub.k]>].sup.2] < [infinity] for every k.

(3) Concerning Lemma 3.1 (iii):

If dom(D) = [l.sup.2] then [[summation].sup.[infinity]].sub.k=1] [c.sub.k][[psi].sub.k] converges unconditionally for every ([c.sub.k]) [member of] [l.sup.2] and [PSI] is a Bessel sequence for H (see [10, Cor. 3.2.4 and 3.2.5]). Now one could wonder whether this is pointwise true for an arbitrary sequence, i.e. whether c [member of] dom(D) implies that [[summation].sup.[infinity].sub.k=1] [c.sub.k][[psi].sub.k] is already unconditionally convergent. This is not true in general, as shown by the following easy example. Consider the non-Bessel sequence [PSI] = ([e.sub.1],[e.sub.1], [e.sub.1],...) and ([c.sub.k]) = (1,- 1/2,1/3, -1/4,...) [member of] [l.sup.2]. Then [[summation].sup.[infinity].sub.k=1] [[c.sub.k], [[psi].sub.k] = (ln 2)[e.sub.1]. However, if ([[member of].sub.k]) = (1,-1, 1,-1,...), then the series [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not converge, which by [19, Theor. 2.81 implies that [[summation].sup.[infinity].sub.k=1] [c.sub.k][[psi].sub.k] is conditionally convergent.

(4) Concerning Lemma 3.1(iv), (v):

The assumption in (iv) and (v) is clearly satisfied when [PSI] is a Bessel sequence for H (in which case dom(C) = H and dom(G) = [l.sup.2]). However, the validity of [[summation].sup.[infinity].sub.l=1] [[absolute value of <[[psi].sub.l], [[psi].sub.k]>].sup.2] < [infinity] for every k [member of] N does not require [PSI] to be a Bessel sequence for H, consider for example [PSI] = ([e.sub.1], 2[e.sub.2], 3[e.sub.3],...).

(5) Concerning Lemma 3.1(vi):

For frames, the assumption in (vi) is equivalent to the Hilbert space being finite dimensional [2]. This is not necessarily valid any more for other sequences, consider for example [PSI] = ([e.sub.1], [e.sub.2]/2, [e.sub.3]/3, [e.sub.4]/4,...), it satisfies (vi) and H is infinite dimensional.

3.2 Connection between the associated operators

Having defined the operators associated to an arbitrary sequence, we will now investigate the relationships among them.

Proposition 3.3 For an arbitrary sequence [PSI], the following statements hold.

(i) C = D* (1).

(ii) [9, Prop. 4.6] If C is densely defined, then D [subset or equal to] C*.

(iii) S = DC.

(iv) CD [subset or equal to] G.

(v) [7, Lemma 3.1] C is closed.

(vi) S is symmetric (2), but not necessarily densely defined; it is positive on dom(S) and positive definite on [bar.span] {[PSI]} [intersection] dom(S).

(vii) If C is densely defined, then S is closable.

(viii) D is closable if and only if C is densely defined.

(ix) D is closed if and only if C is densely defined and D = C*.

(x) If D is closed, then G is densely defined.

(xi) If [[summation].sup.[infinity].sub.l=1] [[absolute value of <[[psi].sub.l], [[psi].sub.k]>].sup.2] < [infinity] for every k, then G is closed.

Proof: (i) First we show that dom(D*) [subset or equal to] dom(C). Fix f [member of] dom(D*). This means that D : dom(D) [right arrow] C, D(c) := (Dc, f}, is a bounded functional. Since dom(D) is dense in [l.sup.2], there is a unique bounded extension [D.sub.0] : [l.sup.2] [right arrow] C. For c = [([c.sub.k]).sup.[infinity]].sub.k=1] [member of] [l.sup.2], denote

[c.sub.N] = ([c.sub.1],[c.sub.2],... ,[c.sub.N],0,0,...) [member of] [c.sub.00] [subset or equal to] dom(D).

Clearly, [c.sub.N] [right arrow] c in [l.sup.2]-norm as N [right arrow] [infinity]. Hence, [D.sub.0][c.sub.N] [right arrow] [D.sub.0]c as N [right arrow] [infinity]. Therefore, [D.sub.0][c.sub.N] [[summation].sup.N.sub.k=1] [c.sub.k] [<[[psi].sub.k], f>.sub.H] converges as N [right arrow] [infinity] for every c [member of] [l.sup.2]. Now Lemma 2.2 implies that ([<[[psi].sub.k], f>.sub.H]) [member of] [l.sup.2], which proves that f belongs to dom(C).

Now we show that dom(C) [subset or equal to] dom(D*). Let f [member of] dom(C). For every c = [([c.sub.k]).sup.[infinity].sub.k=1] [member of]dom(D),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Therefore, the functional c [right arrow] [(Dc, f).sub.H] is bounded on dom(D) by [parallel]Cf[parallel]. Hence, f [member of] dom(D*).

Furthermore, (2) implies that D*f = Cf due to the uniqueness of D*.

(iii) Assume that f [member of] dom(S). Then f [member of] dom(C) (see Lemma 3.1(i.1)). Furthermore, Cf [member of] dom(D). Therefore, dom(S) [subset or equal to] dom(DC). The converse inclusion dom(DC) [subset or equal to] dom(S) is obvious. Now it is clear that S = DC.

(iv) Let c = [([c.sub.k]).sup.[infinity]].sub.k=1] [member of] dom(CD). Then [[summation].sup.[infinity].sub.l=1] [c.sub.l] <[[psi].sub.l], [[psi].sub.k]> converges for every k and it is equal to <[D.sub.c], [[psi].sub.k]>. Furthermore, [D.sub.c] [member of] dom(C), which implies that c [member of] dom(G) and Gc = CDc.

(v) Since the statement is given in [7] without proof, for the sake of completeness we refer to [28, Lemma 3.1] for a proof. Note that [28, Lemma 3.1] concerns sequences satisfying the lower p-frame condition, but the proof that C is closed does not use the validity of the lower p-frame condition.

(vi) Let f, g [member of] dom(S). Then <Sf, g> = [[summation].sup.[infinity].sub.k=1], <f, [[psi].sub.k]> <[[psi].sub.k], g> = <f, Sg>, which means that S is symmetric. An example of a sequence [PSI] with S being non-densely defined can be seen in Remark 3.2(1). Further, <Sf, f> = [[summation].sup.[infinity].sub.k=1] [[absolute value of <f, [[psi].sub.k]>].sup.2] [greater than or equal to] for every f [member of] dom(S). If [[summation].sup.[infinity].sub.k=1] [[absolute value of <f, [[psi].sub.k]>].sup.2] = 0, then f [member of] (span {[PSI]}).sup.[perpendicular to]] which implies that S is positive definite on [bar.span]{[PSI]} [intesection] dom(S).

(vii) By (v), C is closed. Assume that C is densely defined. Then Proposition 2.3(e) implies that C*C is self-adjoint and thus, closed. By (ii) and (iii), we have that S = DC [subset or equal to] C*C, which implies that S is closable.

(viii) and (ix) follow from (i), Lemma 3.1(iii) and Prop. 2.3(b),(c) applied with F = D.

(x) Let D be closed. By Lemma 3.1(iii) and Proposition 2.3(e), it follows that D*D is self-adjoint, in particular, densely defined. Using (i) and (iv), it follows that D*D [subset or equal to] G, which implies that G is also densely defined.

(xi) Assume that [[summation].sup.[infinity].sub.l=1] [[absolute value of <[[psi].sub.l], [[psi].sub.k]>].sup.2] for every k [member of] N. Consider a sequence of elements [c.sup.n] [member of] dom(G), say [c.sup.n] = [([[c.sup.n].sub.k]).sup.[infinity]].sub.k=1], converging in [l.sup.2] to some element c = [([c.sub.k]).sup.[infinity]].sub.k=1] and such that G[c.sup.n] converges to d = [([d.sub.k]).sup.[infinity]].sub.k=1] in [l.sup.2]. Fix k [member of] N and [epsilon] > 0. Let N [member of] N be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [[summation].sup.[infinity].sub.l=1] [c.sup.N.sub.l] <[[psi].sub.l], [[psi].sub.k]> converges, there exists [N.sub.1] so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, for every i, j > [N.sub.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, the series [[summation].sup.[infinity]].sub.l=1] [c.sub.l] <[[psi].sub.l], [[psi].sub.k]> converges. Furthermore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, since G[c.sup.n] [right arrow] d in [l.sup.2] as n [right arrow] [infinity] and since convergence in [l.sup.2] implies convergence by coordinates, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [[summation].sup.[infinity]].sub.l=1] [c.sub.l] <[[psi].sub.l], [[psi].sub.k]> = [d.sub.k] for every k [member of] N, which implies that c [member of] dom(G) and Gc = d.

Remark 3.4 (concerning Prop. 3.3(ii)): If C is densely defined, in general one should not expect the validity of D = C*. A counterexample can be found in [9] after Corollary 4.7. When [PSI] is a Bessel sequence for H, then D = C* (see, e.g., [10, Theor. 3.2.3 and Lemma 3.2.1]).

Note that Proposition 3.3(ix) extends [9, Corollary 4.7].

3.3 Kernels

Lemma 3.5 For an arbitary sequence [PSI], the following statements hold.

(i) ker(S) : ker(C) = [([bar.span]{[PSI]}).sup.[perpendicular]] = [(ran(D)).sup.[perpendicular]]

(ii) ker(D) [subset or equal to] [[bar.ran(C)].sup.[perpendicular]].

Proof: (i) Assume that f [member of] ker(S). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which implies that f [member of] ker(C). It is obvious that ker(C) [subset or equal to] ker(S). The equality ker(C) = [[bar.span]{[PSI]}.sup.[perpendicular]] is also obvious. The equality ker(C): [(ran(D)).sup.[perpendicular]] follows from Lemma 3.1(iii), Proposition 3.3(i) and Proposition 2.3(d) with F = D.

(ii) Follows easily using Prop. 3.3(i).

3.4 Ranges

Lemma 3.6 For an arbitrary sequence [PSI], the following statements hold.

(i) ran(S) [subset or equal to] ran(D).

(ii) If ran(S) : ran(D), then [bar.ran(C)] [intersection] dom(D) = ran(C) [intersection] dom(D).

(iii.1) Let [PSI] be a Bessel sequence for H. Then ran(S) [subset or equal to] ran(D) [subset or equal to] [bar.ran(S)] (with dense inclusion).

(iii.2) Let [PSI] be a Bessel sequence for H. Then ran(S) = ran(D) if and only if [PSI] is a frame sequence.

Proof: (i) follows from Prop. 3.3(iii).

(ii) Assume that ran(S) = ran(D). Let c [member of] [bar.ran(C)] [intersection] dom(D). Since Dc [member of] ran(D) = ran(S), there exists f [member of] dom(S) [subset or equal to] dom(C) so that Dc = DCf. Since [D.sub.[bar.ran(C)][intersection]dom(D)] is injective (see Lemma 3.5(ii)), it follows that e = C f [member of] ran(C). Therefore, [bar.ran(C)] [intersection] dom(D) [subset or equal to] ran(C) [intersection] dom(D). The inverse inclusion is obvious.

(iii.1) Let f [member of] ran(D). Since [PSI] is a Bessel sequence for H, it follows that ker D = ker(C*) = [[bar.ran(C)].sup.[perpendicular]], which implies that f = Dc for some c [member of] [bar.ran(C)]. Therefore, f = [lim.sub.n[right] arrow][infinity] DC[f.sub.n] = [lim.sub.n[right] arrow][infinity] S[f.sub.n] for some [f.sub.n] [member of] H, n [member of] N, such that c = [lim.sub.n[right] arrow][infinity] C[f.sub.n]. This completes the proof.

(iii.2) Assume that ran(S) = ran(D). Since [PSI] is a Bessel sequence for H, it follows that dom(D) = H and (ii) implies that ran(C) is closed. Now [10, Cor. 5.5.3] (see Prop. 4.1(b)) implies that [PSI] is a frame sequence. Conversely, if [PSI] is a frame sequence, then ran(S) = ran(D) = [bar.[span]]{[PSI]} (see, e.g., [10, Lemma 5.1.5 and Theor. 5.5.1]).

4 Classification of General Sequences

In this section, we turn to the central topic of the paper, namely, the classification of arbitrary sequences. We consider two different methods of classification, first in terms of the associated operators, then via orthonormal bases.

Note that throughout this section, we use a rather unconventional numbering technique to increase the comparability for the convenience of the reader. The items (a), (b), ... refer always to the same class of sequences, Bessel sequences, frame sequences, .... Therefore, if no result is available for a given class, we simply omit the corresponding item.

4.1 Classification by the associated operators

In the next four propositions, we proceed with the classification of arbitrary sequences in terms of the operators C, D, S, and G.

Proposition 4.1 Given a sequence [PSI], the following statements hold.

(a1) [19, Theor. 7.2(a)] [PSI] is a Bessel sequence for H if and only if dom(C) = H.

(a2) [PSI] is a Bessel sequence for H with bound B if and only if dom(C) = H and C is bounded with [parallel]C[parallel] [less than or equal to] [square root of B].

(b) [10, Cor. 5.5.3] [PSI] is a frame sequence if and only if dom(C) = H and ran(C) is closed.

(c) [10, Cor. 5.5.3] [PSI] is a frame for H if and only if dom(C) = H ran(C) is closed and C is injective.

(d) [PHI] is a Riesz basis for H if and only if dom(C) = H and C is bijective.

(e) [7, Lemma 3.1] [PSI] is a lower frame sequence for H if and only if C is injective and ran(C) is closed.

(f) [30, Ch.4 Sec.2] [PSI] is a Riesz-Fischer sequence if and only if C is surjective.

(g) [PSI] is complete in H if and only if C is injective.

Proof: (a2) Having (a1) proved, then (a2) is obvious.

(d) By [10, Theor. 5.4.1] and [16, Prop. 5.1.5], [PSI] is a Riesz basis for H if and only if [PSI] is a frame for H and C is surjective. The rest follows from (c).

(g) follows from Lemma 3.50).

Proposition 4.2 Given a sequence [PSI], the following statements hold.

(al) [10, Cor. 3.2.4 and Theor. 3.2.3] [PSI] is a Bessel sequence if and only if dom(D) = [l.sup.2] .

(a2) [10, Theor. 3.2.3] [PSI] is a Bessel sequence with bound B if and only if dom(D) = [l.sup.2] and D is bounded with [parallel]D[parallel] [less than or equal to] [square root of B].

(b1) [10, Cor. 5.5.2] [PSI] is a frame sequence if and only if dom(D) = [l.sup.2] and ran(D) is closed.

(b2) [PSI] is a frame sequence if and only if ran(D) is closed and ran(D) [subset or equal to] dom(C).

(b3) [PSI] is a frame sequence 'if and only if dom(D) = [l.sup.2] and ran(D) = ran(S).

(c) [10, Theor. 5.5.1] [PSI] is a flame if and only if dom(D) = [l.sup.2] and D is surjective.

(d) [PSI] is a Riesz basis for H if and only if dom(D) = [l.sup.2] and D is bijective.

(e) [PSI] is a lower frame sequence for H if and only if ran(D) is dense in H and ran(D*) is closed.

(f) [PSI] is a Riesz-Fischer sequence if and only if D is injective and [D.sup.-1] is bounded on ran(D).

(g) [PSI] is complete in H if and only if ran(D) is dense in H.

Proof: (b2) Assume that ran(D) is closed and ran(D) [subset or equal to] dom(C). Then H = ran(D) [direct sum] [(ran(D)).sup.[perpendicular]] [subset or equal to] dom(C), which by Proposition 4.1(a1) implies that [PSI] is a Bessel sequence for H. Now apply (a1) and (b1).

Conversely, assume that [PSI] is a frame sequence. By (b1), ran(D) is closed, and clearly, dom(C) = H [contains or equal to] ran(D).

(b3) follows from Lemma 3.6(iii.2) and (a1).

(d) By [10, Theor. 5.4.1 and 6.1.1], [PSI] is a Riesz basis for H if and only if [PSI] is a frame for H and D is injective. The rest follows from (c).

(e) By Propositions 4.1(e) and 3.3(i), [PSI] is a lower frame sequence for H if and only if D* is injective and ran(D*) is closed. Since D is densely defined, Proposition 2.3(d) completes the proof.

(f) is clear (see, e.g., [20, V.4.3 Theorem 2 and the Remark after that]).

(g) follows from Lemma 3.5(i).

Note that (f) (a result about the Riesz-Fischer sequences) is missing in Proposition 4.3. The operator S does not distinguish between a frame which is a Riesz-Fischer sequence and a frame which is not. For example, an orthonormal basis and a Parseval frame have the same frame operator, the identity. This can also be seen in Proposition 4.3(d), where the properties of ([S.sup.-1][[psi].sub.k]) have to be taken into account.

Proposition 4.3 Given a sequence [PSI], the following statements hold.

(al) [PSI] is a Bessel sequence for H if and only if dom(S) = H.

(a2) [PSI] is a Bessel sequence for H with bound B if and only if dom(S) = H and S is bounded with [parallel]S[parallel] [less than or equal to] B.

(bl) [PSI] is a frame sequence if and only 'if dom (S) = H and ran(S) is closed.

(b2) [PSI] is a frame sequence if and only if dom (S): H and ran(S) = [bar.span]{[PSI]}.

(c1) [8, Theorem 2.1] [PSI] is a frame for H if and only if dom (S)= H and S is surjective.

(c2) [PSI] is a frame for H if and only if dom (S) = H and S is bijective.

(d) [PSI] is a Riesz basis for H if and only if dom(S) = H, S is bijective and [([S.sup.-1][[psi].sub.k]).sup.[infinity].sub.k=1] is biorthogonal to [PSI].

(e) [PSI] is a lower frame sequence for H if and only if S is injective and ran(C) is closed.

(g) [PSI] is complete in [PSI] if and only if S is injective.

Proof: (al) Let dom(S) = H. The validity of the upper frame inequality follows from the first part of the proof in [8, Theorem 2.1]. For the sake of completeness, we sketch a proof. If dom(S) = [PSI], then [summation] [[absolute value of <f, [[psi].sub.k]>].sup.2] = <Sf, f> [infinity] for every f [member of] H, which implies that dom(C) = H and the rest follows from Proposition 4.1(al). For the other direction, see the text after Example 5.1.4 in [10, See. 5.1].

(a2) See (a1); the boundedness of S can be found, e.g., in the proof of [10, Lemma 5.1.5].

(b1) Assume that dom(S) = H and ran(S) is closed. By (a1), [PSI] is a Bessel sequence for H. Then Lemma 3.6(iii.1) implies that ran(D) is closed, which by Proposition 4.2(b1) implies that [PSI] is a frame sequence.

Conversely, if [PSI] is a frame sequence, the conclusion follows from [8, Theorem 2.1] (see (cl)).

(b2) follows from [8, Theorem 2.1] (see (c1)).

(c2) follows from (c1) and [10, Lemma 5.1.5].

(d) Use (c2) and the fact that [PSI] is a Riesz basis for H if and only if [PSI] is a frame for H and [([S.sup.-1][[psi].sub.k]).sup.[infinity].sub.k=1] is biorthogonal to [PSI] [10, Theor. 5.4.1 and 6.1.1].

(e) By Lemma 3.5(i), S is injective if and only if C is injective. The rest follows from Proposition 4.1(e).

(g) follows from Lemma 3.5(i).

Proposition 4.4 For a sequence [PSI], the following statements hold.

(a1) [PSI] is a Bessel sequence for H if and only if dom(G) = [l.sup.2].

(a2) [10, Lemma 3.5.1] [PSI] is a Bessel sequence for H with bound B if and only if dom(G) : [l.sup.2] and G is bounded with [parallel]G[parallel] [less than or equal to] B.

(a3) If [PSI] is a Bessel sequence for H, then [G|.sub.ran(C)] is an injeetive operator from ran(C) into ran(C) and ran([G|.sub.ran(C)]) is dense in ran(C).

(b) [PSI] is a frame sequence if and only if dom(G) = [l.sup.2] and [G|.sub.ran(C)] is a bounded operator from ran(C) onto ran(C) with bounded inverse.

(c) [PSI] is a frame for H if and only if [PSI] is complete in H, dom(G) = [l.sup.2] and [G|.sub.ran(C)] is a bounded operator from ran(C) onto ran(C) with bounded inverse.

(d) [10, Theor. 3.6.6] [PSI] is a Riesz basis for H if and only if [PSI] is complete in H and G is a bounded invertible operator on H.

(f) [30, Ch.4 Sec.2] [PSI] is a Riesz-Fischer sequence with bound A if and only if the "sections" [G.sub.n] of G satisfy the inequality A [parallel]c[parallel] [less than or equal to] [parallel][G.sub.n]c[parallel] for all finite sequences and all n, where the "section" [G.sub.n] is the matrix [(<[[psi].sub.l], [[psi].sub.k]>).sup.n.sub.l,k=1].

Proof: (a1) Let dom(G) = [l.sup.2]. Then G admits a matrix representation via the orthonormal basis [([[delta].sub.k]).sup.[infinity].sub.k=1] of [l.sup.2] and thus, G is bounded [I, Sec. 29 Theor. 1]. (3) Since [1] is in Russian, for convenience of the reader we add a sketch of a proof. Let n [member of] N. Consider the operator [G.sub.n] defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then dom([G.sub.n]) : [l.sup.2] and [G.sub.n] is bounded. Therefore, by the Banach-Steinhaus theorem, G is bounded. Now the rest follows from [10, Lemma 3.5.1] (see (a2)).

(a3) follows from [10, Lemma 3.5.2] and Proposition 3.3(i).

(b) If [PSI] is a frame sequence, the statement follows from [10, Prop. 5.2.2] and Proposition 3.3(i). Conversely, assume that dom(G) = [l.sup.2] and [G|.sub.ran(C)] is a bounded operator from ran(C) onto ran(C) with bounded inverse. Then [PSI], is a Bessel sequence for H and thus G = CD. Consider the subspace [bar.span]{[PSI]} of H and the operator [C.sub.1] = [C.sub.[bar.span]{[PSI]}]. By Lemma 3.5(i), [C.sub.1] is a bijection from [bar.span]{[PSI]} onto ran(C). Since ran(D) [subset or equal to] [bar.span]{[PSI]}, for every c [member of] ran(C) one has [G|.sub.ran](C)C = C Dc = [C.sub.1][D|.sub.ran](C)c and thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies that [C.sup.-1.sub.1] is bounded. Thus, the bounded injective operator [C.sup.1] has a bounded inverse on ran([C.sub.1]), which implies that [C.sub.1] has closed range [29, Cot. IV.3.6] (for a reference in English, see [13, Ex. VI.9.15(i)]). Therefore, by Proposition 4.1(c), [PSI] is a frame for [bar.span]{[PSI]}.

(c) follows from (b).

Note that again some items are missing, as in Proposition 4.3. It is impossible to have some connection of properties of the Gram matrix to completeness. For example, the standard orthonormal basis ([e.sub.1], [e.sub.2], [e.sub.3], ... ) as well as ([e.sub.2], [e.sub.3], [e.sub.4],. ...) have the same Gram matrix. The problem concerning sequences that satisfy the lower frame condition is still open.

4.2 Operators that preserve the sequence properties

Proposition 4.5 Given a sequence [PSI], the following statements hold.

(a) Let [PSI] be a Bessel sequence for H with bound B and let F: span{[PSI]} [right arrow] H be a bounded operator. Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a Bessel sequence for H with bound B [[parallel]F[parallel].sup.2].

(b) Let [PSI] be a frame sequence ,with bounds A, B and let F : [bar.span]{[PSI]}[right arrow] [bar.span]{[PSI]} be a bounded surjective operator. Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a frame sequence with frame bounds A[[parallel][F.sup.[dagger]][parallel].sup.-2] and B[[parallel]F[parallel].sup.2], where [F.sup.[dagger]]] (4) denotes the pseudo-inverse of F.

(c) [10, Cor. 5.3.2] Let [PSI] be a frame for H with bounds A, B and let F: H [right arrow] H be a bounded surjective operator. Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a frame for H with bounds A[[parallel][F.sup.[dagger]][parallel].sup.-2] and B[[parallel]F[parallel].sup.2], where [F.sup.[dagger]] denotes the pseudo-inverse of F.

(d) Let [PSI] be a Riesz basis for H with bounds A,B and let F : H [right arrow] H be a bounded bijective operator. Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a Riesz basis for H with bounds A[[parallel][F.sup.-1]][parallel].sup.-2] and B[[parallel]F[parallel].sup.2].

(e) Let [PSI] be a lower frame sequence for H with bound A and let the operator F : [bar.span{[PSI]}] [right arrow] H be bounded and such that F* has a bounded inverse on ran(F*) (equivalently, F be bounded and surjective). Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a lower frame sequence for H with bound A[[parallel][(F*).sup.-1][parallel].sup.-2]

(f) Let [PSI] be a Riesz-Fischer sequence with bound A and let the operator F : span{[PSI]} [right arrow] H have a bounded inverse [F.sup.-1] : ran(F) [right arrow] H with bound K. Then [MATHEMATICAL [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a Riesz-Fischer sequence with bound [AK.sup.-2].

(g) Let [PSI] be complete in H and let F: span{[PSI]} [right arroow] H be bounded with ran(F) being dense in H. Then [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is complete in H.

Proof: (a) Let [bar.F] denote the bounded extension of F on [bar.span]{[PSI]} without increasing the norm. For every f [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) follows from [10, Cor. 5.3.2] (see (c)).

(d) By (c), [([F[[psi].sub.k]).sup.[infinity].sub.k=1] is a frame for H and thus, complete in H (see Proposition 4.2(c)). Furthermore, for every finite sequence ([c.sub.k]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and similarly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(e) A bounded operator from a Banach space into a Banach space is surjective if and only if its adjoint has a bounded inverse on the range of the adjoint (see, e.g., [27, Theorem 4.15] and [20, V.4.3 Theorem 2 and the Remark after thatl). For every f [member of] H,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(f) As in (d), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every finite sequence ([c.sub.k]).

(g) Let [PSI] be complete in H and let [bar.F] denote the bounded extension of F on [bar.span]{[PSI]} = H. Assume that f [member of] H and (f, F[[psi].sub.k]) = 0, [for all]k [member of] N. Then [bar.F]*f = 0. Since ran([bar.F]) is dense in H, it follows that [bar.F]* is injective (see Proposition 2.3(d)). Therefore, f = 0.

4.3 Classification with Orthonormal Bases

Another way of classifying sequences is to consider them as images of orthonormal bases under specific classes of operators. This we do in the present subsection. From now on, [D.sub.[PSI]] denotes the synthesis operator for [PSI] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the analysis operator for [([e.sub.k]).sup.[infinity].sub.k=1]

Proposition 4.6 Let [([e.sub.k]).sup.[infinity].sub.k=1] be an orthonormal basis for H.

(a) The Bessel sequences for H are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1] where V: H [right arrow] H is a bounded operator.

(b) The frame sequences for H are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1] where V: H [right arrow] H is a bounded operator with closed range.

(c) [10, Theor. 5.5.5] The frames for H are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1], where V: H [right arrow] H is a bounded and surjective operator.

(d) [10, Def. 3.6.1 and Theor. 3.6.6] The Riesz bases for H are precisely the sequences [(V[e.sub.k]).sup.[infinity].sub.k=1] where V: H [right arrow] H is a bounded bijective operator.

(e) The lower frame sequences for H are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1], where V: dom(V) [right arrow] H is a densely defined operator such that [e.sub.k] [member of] dom(V), [for all]k [member of] N, V* is injective with bounded inverse on ran(V*), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(f) [7, Prop. 2.3] The Riesz-Fischer sequences are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1], where V is an operator having all [e.sub.k] in its domain and which has a bounded inverse [V.sup.-1] : ran(V) [right arrow] H.

(g) The complete sequences are precisely the families [(V[e.sub.k]).sup.[infinity].sub.k=1], where V: dom(V) [right arrow] H is a densely defined operator such that [e.sub.k] [member of] dom(V), [for all]k [member of] N, ran(V) is dense in H (equivalently, the adjoint V* is injective) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: (a) If V: H [right arrow] H is a bounded operator, the conclusion follows from Proposition 4.5(a).

Conversely, assume that [PSI] is a Bessel sequence for H. Then the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) If V: H [right arrow] H is a bounded operator with closed range, then [10, Prop. 5.3.1] implies that [(V[e.sub.k]).sup.[infinity].sub.k=1] is a frame sequence.

Conversely, assume that [PSI] is a frame sequence. Consider the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bounded bijection (see Proposition 4.1(d)) and [D.sub.[PSI]], is a bounded operator with closed range (see Proposition 4.2(bl)), it follows that V is bounded and ran(V) is closed. Clearly, [PSI] [(V[e.sub.k]).sup.[infinity].sub.k=1]

(e) Let [PSI] be a lower frame sequence for H. Consider the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is clear that dom(V) contains all [e.sub.k], k [member of] N, (and thus, V is densely defined) and [(V[e.sub.k]).sup.[infinity].sub.k=1]. Using the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bounded bijection of H onto [l.sup.2] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] maps bijectively dora(V) onto dom(D), it is not difficult to see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using Proposition 3.3(i), it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now Propositions 4.1(e) and 4.2(d) imply that V* is injective. Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is bounded on ran (V*). Now assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, assume that V : dom(V) [right arrow] H satisfies the conditions in the statement of (e). Let f [member of] H be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. First we prove that f belongs to dom(V*). Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be an arbitrary element of dora(V). For every [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] belongs to dom(V) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the assumptions, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, taking the limit as n [right arrow] [infinity] in the above inequalities, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, f [member of] dom(V*) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(f) Since the statement in [7, Prop. 2.3] is given without proof, for the sake of completeness we add a proof here. One of the directions follows from Proposition 4.5(f). For the other direction, assume that [PSI] is a Riesz-Fischer sequence with bound A. Consider then the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then the domain of V contains all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every f [member of] dom(V). Therefore, V is injective with bounded inverse on ran(V).

(g) Let [PSI] be complete in H. Consider the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In a similar way as in (e), it follows that V has the desired properties.

Conversely, assume that V: dom(V) [right arrow] H satisfies the conditions in the statement of (g). Let f [member of] H be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] First we prove that f belongs to dom(V*). Every finite sum [SIGMA][c.sub.k][e.sub.k] belongs to dom(V) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] By the above, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] By assumption, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Hence, {Vh, f} = 0, [for all]h [member of] dom(V), which implies that f [member of] dom(V*). Therefore, the assumptions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] imply that {V*f, ek} = 0, [for all]k [member of] N, which implies that V*f = 0 and the injectivity of V* implies that f = 0. This ends the proof that [(V[e.sub.k]).sup.[infinity].sub.k=1] is complete in H.

Note that, in each of the statements in Proposition 4.6, the orthonormal basis can be replaced by a Riesz basis, in view of Proposition 4.6(d).

ACKNOWLEDGEMENT

The authors would like to thank M. A. El-Gebeily as well as the anonymous reviewers for helpful comments and suggestions. This work was partly supported by the WWTF project MULAC ('Frame Multipliers: Theory and Application in Acoustics', MA07-025). The first two authors are thankful for the hospitality of the Institut de Recherche en Mathematique et Physique, Universite catholique de Louvain. The second author is grateful for the support from the MULAC-project. She also thanks the University of Architecture, Civil Engineering and Geodesy, and the Department of Mathematics of UACEG for supporting the present research.

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Peter Balazs a) Acoustics Research Institute, Wohilebengasse 12-14, Vienna A-1040, Austria Peter.Balazs@oeaw.ac.at

Diana T. Stoeva a),b) b) Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, Blvd Hristo Smirnenski I, 1046 Sofia, Bulgaria std73std@yahoo.com

Jean-Pierre Antoine Institut de Recherche en Mathematique et Physique, Universit6 catholique de Louvain, B - 1348 Louvain-la-Neuve, Belgium Jean-Pierre.Antoine@uclouvain.be

(1) This result was obtained independently in [15].

(2) The terminology is not uniform in the literature. We use the definition of a symmetric operator given by [14, Def. XII.1.7], [27, Def. 13.3], namely, S is symmetric if <Sf, g> = (f, Sg}, [for all] f, g [member of] dom(S), without the assumption of a dense domain.

(3) For a matrix representation using frames, we refer to [3].

(4) For the pseudo-inverse of a bounded operator with closed range see, e.g., [10, App. A.7].