# Subnormal weighted shifts on directed trees and composition operators in [L.sup.2]-spaces with nondensely defined powers.

1. Introduction

The question of when powers of a closed densely defined linear operator are densely defined has attracted considerable attention. In 1940 Naimark gave a surprising example of a closed symmetric operator whose square has trivial domain (see [1]; see also [2] for a different construction). More than four decades later, Schmiidgen discovered another pathological behaviour of domains of powers of symmetric operators (cf. [3]). It is well known that symmetric operators are subnormal (cf. [4, Theorem 1 in Appendix I.2]). Hence, closed subnormal operators may have nondensely defined powers. In turn, quasinormal operators, which are subnormal as well (see [5, 6]), have all powers densely defined (cf. [6]). In the present paper we discuss the above question in the context of subnormal weighted shifts on directed trees and subnormal composition operators in [L.sup.2]-spaces (over o-finite measure spaces).

As recently shown (cf. [7, Proposition 3.1]), formally normal (in particular symmetric) weighted shifts on directed trees are automatically bounded and normal (in general, formally normal operators are not subnormal, cf. [8]). The same applies to symmetric composition operators in [L.sup.2]-spaces (cf. [9, Proposition B.1]). Formally normal composition operators in [L.sup.2]-spaces, which may be unbounded (see [9, Appendix C]), are still normal (cf. [10, Theorem 9.4]). As a consequence, all powers of such operators are densely defined (see, e.g., [11, Corollary 5.28]).

The above discussion suggests the question of whether for every positive integer n there exists a subnormal weighted shift on a directed tree whose nth power is densely defined while its (n + 1)th power is not. A similar question can be asked for composition operators in [L.sup.2]-spaces. To answer both of them, we proceed as follows. First, by applying a recently established criterion for subnormality of weighted composition operators in [L.sup.2]-spaces which makes no appeal to density of [C.sup.[infinity]]-vectors (see Theorem 1), we show that a densely defined weighted shift on a directed tree which admits a consistent system of probability measures (i.e., a system {[[mu].sub.v]}.sub.v[member of]V] Borel probability measures on [R.SUB.+] which satisfies (6)) is subnormal and, what is more, its nth power is densely defined if and only if all moments of these measures up to degree n are finite (cf. Theorem 3). The particular case of directed trees with one branching vertex is examined in Theorem 5 and Corollary 6. Using these two results, we answer both questions in the affirmative (see Example 1 and Remark 8). It is worth pointing out that though directed trees with one branching vertex have simple structure, they provide many examples which are important in operator theory (see e.g., [12,13]).

Now we introduce some notation and terminology. In what follows, Z, [Z.sub.+], N, [R.sub.+], and C stand for the sets of integers, nonnegative integers, positive integers, nonnegative real numbers and complex numbers, respectively. Set [[bar.R].sub.+] = [R.sub.+] [union] ([infinity]. We write B([R.sub.+]) for the o-algebra of all Borel subsets of [R.sub.+]. Given t e [R.SUB.+], we denote by St the Borel probability measure on [R.sub.+] concentrated on {t}.

The domain of an operator A in a complex Hilbert space H is denoted by D(A) (all operators considered in this paper are linear). Set [D[infinity]](A) = [[intersection].sup.[infinity].sub.n=0] D([A.sup.n]). Recall that a closed densely defined operator A in H is said to be normal if [AA.sup.*] = [A.sup.*] A (see [11,14,15] for more on this class of operators). We say that a densely defined operator A in H is subnormal if there exist a complex Hilbert space K and a normal operator N in K such that H c K (isometric embedding) and Ah = Nh for all h [member of] D(S). We refer the reader to [6, 16-19] for the foundations of the theory of bounded and unbounded subnormal operators, respectively.

2. Weighted Composition Operators

Assume that (X, A, v) is a [sigma]-finite measure space, w : X [right arrow] C is an A-measurable function, and [phi] : X [right arrow] X is an A-measurable mapping. Define the o-finite measure [v.sub.w] : A [right arrow] [R.sub.+] by [v.sub.w]([DELTA]) = [[integral].sub.[DELTA]] [[absolute value of w].sup.2] dv for {DELTA] [member of] A. Let [v.sub.w] [omicron] [[phi].sup.-1] : A [right arrow] [R.sub.+] be the measure given by [v.sub.w] [omicron] [[phi].sup.-1] ({DELTA]) = [v.sub.w]([[phi].sup.-1] ({DELTA])) for [DELTA] [member of] A. Assume that [v.sub.w] [omicron] [[phi].sup.-1] is absolutely continuous with respect to v. By the Radon-Nikodym theorem (cf. [20, Theorem 2.2.1]), there exists a unique (up to a.e. [v] equivalence) A-measurable function h = [h.sub.[phi].w] : X [right arrow] [R.sub.+] such that

[v.sub.w] [omicron] [[phi].sup.-1] ([DELTA]) = [[integral].sub.[DELTA]] h dv, [DELTA] [member of] A. (1)

Then the operator C = [C.sub.[phi],w] in [L.sup.2](v), given by

D (C) = {f [member of] [L.sup.2] (v): w(f [omicron] [phi]) [member of] [L.sup.2] (v)},

C f = w x (f [omicron] [phi]), f [member of] D (C),

is well defined (cf. [21, Proposition 7]). Call C a weighted composition operator. By [21, Proposition 10], C is densely defined if and only if h < [infinity] a.e. [v]; moreover, if this is the case, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-finite and, by the Radon-Nikodym

theorem, for every A-measurable function f : X [right arrow] [R.sub.+] there exists a unique (up to a.e. [v.sub.w]] equivalence) [[phi].sup.-1](A)- measurable function E(f) = [E.sub.[phi],w](f) : X [right arrow] [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

We call E(f) the conditional expectation of f with respect to [[phi].sup.-1](A) (see [21] for more information). A mapping P : X x B([R.sub.+]) [right arrow] [0,1] is called an A-measurable family of probability measures if the set-function P(x, x) is a probability measure for every x [member of] X and the function P(x,[sigma]) is A-measurable for every [sigma] [member of] B([R.sub.+]).

The following criterion (read: a sufficient condition) for subnormality of unbounded weighted composition operators is extracted from [21, Theorem 29].

Theorem 1. If C is densely defined, h > 0 a.e. [[v.sub.w]], and there exists an A-measurable family of probability measures P : Xx B([R.sub.+]) [right arrow] [0,1] such that

E (P (x, [sigma])) (x) = [[integral].sub.[sigma]] tP ([phi](x), dt)/h ([phi] (x)) for [v.sub.w]-a.e. x [member of] X, [sigma] [member of] B ([R.sub.+]), (CC)

then C is subnormal.

Regarding Theorem 1, recall that if C is subnormal, then h > 0 a.e. [[v.sub.w]] (cf. [21, Corollary 13]).

3. Weighted Shifts on Directed Trees

Let T = (V, E) be a directed tree (V and E stand for the sets of vertices and edges of T, resp.). Set Chi(w) = (v [member of] V: (u, v) [member of] E} for u [member of] V. Denote by par the partial function from V to V which assigns to a vertex u [member of] V its parent par(u) (i.e., a unique v [member of] V such that (v, u) [member of] E). A vertex u [member of] V is called a root of T if u has no parent. A root is unique (provided it exists); we denote it by root. Set [V.sup.[omicron]] = V \ {root} if T has a root and [V.sup.[omicron]] = V otherwise. We say that u [member of] V is a branching vertex of V and write u [member of] [V.sub.<], if Chi(u) consists of at least two vertices. We refer the reader to [12] for all facts about directed trees needed in this paper.

By a weighted shift on T with weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we mean the operator [[S.sub.[lambda]] in [l.sup.2](V) defined by

D ([S.sub.[lambda]]) = {f [member of] [l.sup.2] (V): [[DELTA].sub.T] f [member of] [l.sup.2] (V)},

[S.sub.[lambda]] f = [[DELTA].sub.T] f, f [member of] D ([S.sub.[lambda]]), (4)

where [[DELTA].sub.T] is the mapping defined on functions f : V [right arrow] C via

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

As usual, [l.sup.2](V) is the Hilbert space of square summable complex functions on V with standard inner product. For u [member of] V, we define [e.sub.u] [member of] [l.sup.2](V) to be the characteristic function of the one-point set {u}. Then [{[e.sub.u]].sub.u[member of] V] is an orthonormal basis of [l.sup.2](V).

The following useful lemma is an extension of part (iv) of [13, Theorem 3.2.2].

Lemma 2. Let be a weighted shift on a directed tree T = (V, E) with weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [S.sup.n.sub.[lambda]] is densely defined if and only if [e.sub.u] [member of] D([S.sup.n.sub.[lambda]]) for every u [member of] [V.sub.<].

Proof. In view of [13, Theorem 3.2.2(iv)], [S.sup.n.sub.[lambda]] is densely defined if and only if [e.sub.u] [member of] D([S.sup.n.sub.[lambda]]) for every u [member of] V. Note that if u [member of] V and Chi(u) = {v}, then [e.sub.u] [member of] D([S.sup.n.sub.[lambda]]) and [S.sub.[lambda]] [e.sub.u] = [[lambda].sub.v][e.sub.v], which implies that [e.sub.u] [member of] D([S.sup.n+1.sub.[lambda]]) whenever [e.sub.v] [member of] D([S.sup.n.sub.[lambda]]). In turn, if Chi(u) = [empty set], then clearly [e.sub.u] [member of] [D.sup.[[infinity]([S.sub.[lambda]]). Using the above and an induction argument (related to paths in T), we deduce that [S.sup.n.sub.[lambda]] is densely defined if and only if [e.sub.u] e D([S.sup.n.sub.[lambda]]) for every u [member of] [V.sub.<].

It is worth mentioning that if [V.sub.<] = [empty set], then, by Lemma 2 and [13, Theorem 3.2.2(iv)] (or by the proof of Lemma 2), [D.sup.[infinity]]( [S.sub.[lambda]]) is dense in [l.sup.2](V). In particular, this covers the case of classical weighted shifts and their adjoints.

Now we give a criterion for subnormality of weighted shifts on directed trees. As opposed to [22, Theorem 5.1.1], we do not assume the density of [C.sup.[infinity]]-vectors in the underlying [l.sup.2]-space. Moreover, we do not assume that the underlying directed tree is rootless and leafless, which is required in [9, Theorem 47], and that weights are nonzero. The only restriction we impose is that the directed tree is countably infinite. This is always satisfied if the weighted shift in question is densely defined and has nonzero weights (cf. [12, Proposition 3.1.10]). Here, and later, we adopt the conventions that 0 x [infinity] = [infinity] x 0 = 0, 1/0 = [infinity] and [[summation].sub.v[member of][phi]] [[zeta].sub.v] = 0; we also write [[integral].sup.[infinity].sub.0] in place

of [[integral].sub.+].

Theorem 3. Let [S.sub.[lambda]] be a weighted shift on a countably infinite directed tree T = (V, E) with weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that there exist a system [{[u.sub.v]}.sub.v[member of]v] of Borel probability measures on [R.sub.+] and a system [{[epsilon].sub.v}.sub.v[member of]V] of nonnegative real numbers such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Then the following two assertions hold:

(i) if [S.sub.[lambda]] is densely defined, then [S.sub.[lambda]] is subnormal,

(ii) if n [member of] N, then [S.sup.n.sub.[lambda]] is densely defined if and only if [[integral].sup.[infinity].sub.0] [s.sup.n] d [[muu].sub.u](s) < [infinity] for all u [member of] [V.sub.<].

Proof. (i) Assume that [S.sub.[lambda]] is densely defined. Set X = V and A = [2.sup.V]. Let v : A [right arrow] [R.sub.+] be the counting measure on X (v is [sigma]-finite because V is countable). Define the weight function w : X [right arrow] C and the mapping [phi] : X [right arrow] X by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

Clearly, the measure [v.sub.w] [omicron] [[phi].sup.-1] is absolutely continuous with respect to v and

h (x) = [v.sub.w] ([[phi].sup.-1]({x})) = [v.sub.w] (Chi (x))= [summation over y[member of]Chi(x)] [absolute value of [[lambda].sub.y]].sup.2]

x [member of] X. (8) Thus, by [12, Proposition 3.1.3], h(x) < [infinity] for every x [member of] X. We claim that h > 0 a.e. p[v.sub.w]]. This is the same as to show that if x [member of] [V.sup.[omicron]] and [v.sub.w](Chi(x)) = 0, then [[lambda].sub.x] = 0. Thus, if x [member of] [V.sup.[omicron]] and [v.sub.w](Chi(x)) = 0, then applying (6) to u = x, we deduce that [[mu].sub.x] = [[delta].sub.0]; in turn, applying (6) to u = par(x) with [sigma] = |0}, we get [[lambda].sub.x] = 0, which proves our claim.

Note that X = [[??].sub.x[member of]X] [[phi].sup.-1]{(x)}s (the disjoint union). Hence, the conditional expectation E(f) of a function f : X [right arrow] [R.sub.+] with respect to [[phi].sup.-1] (A) is given by

E (f) (z) = [[integra].sub.Chi(x)] f d [v.sub.w]/h(x), z [member of] [[phi].sup.-1] ({x}), x [member of [X.sub.+], (9)

where [X.sub.+] := {x [member of] X : [v.sub.w](Chi(x)) > 0} (see also (8)); on the remaining part of X we can put E(f) = 0.

Substituting [sigma] = {0} into (6), we see that [[mu].sub.y] ({0}) = 0 for every y [member of] [V.sup.[omicron]] such that [[lambda].sub.y] = 0. Thus, using the standard measure-theoretic argument and (6), we deduce that

[[integral].sub.[sigma]] t d [[mu].sub.x] (t) = [summation over y[member of]Chi(x)] [[absolute value of [[lambda].sub.y].sup.2] [[mu].sub.y] ([sigma]), [sigma][member of] B([R.sub.+], x [member of] X. (10) Set P(x, [sigma]) = [[mu].sub.x]([sigma]) for x [member of] X and [sigma] [member of] B([R.sub.+]). It follows from (9) and (10) that P : X x B([R.sub.+]) [right arrow] [0,1] is a (A- measurable) family of probability measures which fulfils the following equality:

E (P (x, [sigma]))(z) = [[integral].sub.[sigma]] tP([phi] (z), dt)/h([phi] (z)), z [member of] [[phi].sup.-1] ({x}), x [member of [X.sub.+]. (11)

This implies that P satisfies (CC). Hence, by Theorem 1, the weighted composition operator C (see (2)) is subnormal. Since [S.sub.[lambda]] C, assertion (i) is proved.

(ii) It is easily seen that if [mu] is a finite positive Borel measure on [R.sub.+] and [[integral].sup.[infinity].sub.0] [s.sup.n] d [mu](s) < [infinity] for some n [member of] N, then [[integral].sup.[infinity].sub.0] [s.sup.k] d[mu](s) < [infinity] for every k [member of] N such that k [less than or equal to] n. This fact combined with Lemma 2 and [22, Lemmata 2.3.1 (i) and 4.2.2(i)] implies assertion (ii). Remark 4. Assume that [[S.sub.[lambda]] is a densely defined weighted shift on a countably infinite directed tree T = (V, E) with weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A careful inspection of the proof of Theorem 3 reveals that if {[mu]sub.x]}.sub.x[member of]X] (with X = V) is a system of Borel probability measures on [R.sub.+] which satisfies (6), then h > 0 a.e. [[v.sub.w]], the family P defined by P(x, x) = [[mu].sub.x] for x [member of] X satisfies (CC), and [[mu].sub.x] = [[delta].sub.0] for every x [member of] X \ [X.sub.+]. We claim that if h > 0 a.e. [[v.sub.w]] and P : X x B([R.sub.+]) [right arrow] [0,1] is any family of probability measures which satisfies (CC), then the system {[[??].sub.x]}.sub.x[member of]X] of probability measures defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

satisfies (6) with {[[??].sub.x]}.sub.x[member of]X] in place of {[[??].sub.x]}.sub.x[member of]X]. Indeed, (CC) implies (11). Hence, by (9), equality in (10) holds for every x [member of] [X.sub.+] with [[mu].sub.z] = P(z, x) for z [member of] X. This implies via the standard measure-theoretic argument that equality in (6) holds for every u [member of] [X.sub.+]. Since h > 0 a.e. [[v.sub.w]], we deduce that equality in (6) holds for every u [member of] [X.sub.+] with {[[??].sub.x]}.sub.x[member of]X] in place of {[[??].sub.x]}.sub.x[member of]X]. Clearly, this is also the case for u [member of] X \ [X.sub.+]. Thus, our claim is proved.

4. Trees with One Branching Vertex

Theorem 3 will be applied in the case of weighted shifts on leafless directed trees with one branching vertex. First, we recall the models of such trees (see Figure 1). For [eta], [kappa] [member of] [Z.sub.+] [??] ([infinity] with [[eta] [greater than or equal to] 2, we define the directed tree [T.sub.[eta],[kappa]] = ([V.sub.eta], [kappa]], [[E.sub.[eta], [kappa]]) as follows (the symbol "[??]" denotes disjoint union of sets):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where [J.sub.n] = (k [member of] N : k [less than or equal to] n} for n [member of] [Z.sub.+] [??] {[infinity]}. Clearly, [T.sub.[eta], kappa]] is leafless and 0 is its only branching vertex. From now on, we write [[lambda].sub.i,j] instead of the more formal expression [[lambda].sub.(i,j)] whenever (i, j) [member of [V.sub.[eta], [kappa]].

Theorem 5. Let [eta], [kappa] [member of] [Z.sub.+] [??] {[infinity]} be such that [greater than or equal to] 2 and let [S.sub.[lambda]] be a weighted shift on a directed tree [T.sub.[eta], [kappa]] with nonzero weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that there exists a sequence {[[mu].sub.i]}.sup.[eta].sub.i=1] of Borel probability measures on [R.sub.+] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

and that one of the following three disjunctive conditions is satisfied:

(i) [kappa] = 0 and

[[eta].summation over i=1] [absolute value of [[lambda].sub.i,1]].sup.2] [[integral].sup.[infinity].sub.0] 1/s d [[mu].sub.i](s) [less than or equal to] 1, (15)

(ii) 0 < [kappa] < [infinity] and

[[eta].summation over i=1] [absolute value of [[lambda].sub.i,1]].sup.2] [[integral].sup.[infinity].sub.0] 1/s d [[mu].sub.i](s) [less than or equal to] 1, (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

(iii) [kappa] = [infinity] and equalities (16) and (17) are valid.

Then the following two assertions hold:

(a) if [[S.sub.[lambda]] is densely defined, then is subnormal,

(b) if n [member of] N, then [S.sup.sub.[lambda]] is densely defined if and only if

[[eta].summation over i=1] [absolute value of [[lambda].sub.i,1]].sup.2] [[integral].sup.[infinity].sub.0] [s.sup.n-1] d [[mu].sub.i](s) < [infinity], (19)

Proof. As in the proof of [23, Theorem 4.1], we define the system [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of Borel probability measures on [R.sub.+] and verify that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies (6). Hence, assertion (a) is a direct consequence of Theorem 3(i).

(b) Fix n [member of] N. It follows from Theorem 3(ii) that [S.sup.n.sub.[lambda]] is densely defined if and only if [[integral].sup.[infinity].sub.0] [s.sup.n] d [[mu].sub.0](s) < [infinity]. Using the explicit definition of [[mu].sub.0] and applying the standard measure-theoretic argument, we see that

[[integral].sup.[infinity].sub.0] [s.sup.n] d [[mu].sub.0](s) [[eta].summation over i=1] [absolute value of [[lambda].sub.i,1]].sup.2] [[integral].sup.[infinity].sub.0] [s.sup.n-1] d [[mu].sub.i](s). (20) This completes the proof of assertion (b) (the case of n = 1 can also be settled without using the definition of [[mu].sub.0] simply by applying Lemma 2 and [12, Proposition 3.1.3(iii)]). Note that Theorem 5 remains true if its condition (ii) is replaced by the condition (iii) of [23, Theorem 4.1] (see also [23, Lemma 4.2] and its proof).

Corollary 6. Under the assumptions of Theorem 5, if n [member of] N, then the following two assertions are equivalent:

(i) [S.sup.n.sub.[lambda]] is densely defined and [S.sup.n+1].sub.[lambda]] is not,

(ii) the condition (19) holds and [summation.sup.n.sub.i=1] [[lambda].sub.i,1].sup.2]] [[integral].sup.[infinity].sub.0] [s.sup.n-1] d [[mu].sub.i](s) = [infinity].

5. The Example

It follows from [22, Lemma 2.3.1 (i)] that if [[S.sub.[lambda]] is a weighted shift on and [T.sub.[eta],[kappa]] and [eta] < [infinity], then [D.sup.[infinity] ([S.sub.[lambda]]) is dense in [l.sup.2]([V.sub.[eta],[kappa]] (this means that Corollary 6 is interesting only if [eta] = [infinity]). If [eta] = [infinity], the situation is completely different. Using Theorem 5 and Corollary 6, we show that for every n [member of] N and for every [kappa] [member of] [Z.sub.+] [??] {[infinity]}, there exists a subnormal weighted shift [S.sub.[lambda]] on [T.sub.[infinity],[kappa]] such that [S.sup.n.sub.[lambda]] is densely defined and [S.sup.n+1.sub.[lambda]] is not. For this purpose, we adapt [12, Procedure 6.3.1] to the present context. In the original procedure, one starts with a sequence {[[mu].sub.i]}.sup.[infinity].sub.i-1] of Borel probability measures on [R.sub.+] (whose nth moments are finite for every n [member of] Z such that n [greater than or equal to] -([kappa] + 1)) and then constructs a system of nonzero weights [lambda] = ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that satisfies the assumptions of Theorem 5 (in fact, using Lemma 7 below, we can also maintain the condition (19)). However, in general, it is not possible to maintain the condition (ii) of Corollary 6 even if {[[mu].sub.i]}.sup.[infinity].sub.i-1] are measures with two-point supports (this question is not discussed here).

Example 1. Assume that [eta] = m. Consider the measures [[mu].sub.i] = [[delta].sujb.qi] with [q.sub.i] e (0, [infinity]) for i [member of] N. By [12, Notation 6.1.9 and Procedure 6.3.1], [S.sub.[lambda]] B([l.sup.2]([V.sub.[infinity],[kappa])) if and only if Sup{[q.sub.i] : I [member of] N} < [infinity]. Hence, there is no loss of generality in assuming that sup{[q.sub.i] : i [member of] N} = [infinity]. To cover all possible choices of [kappa] [member of] [Z.sub.+] [??] {[infinity]}, we look for a system of nonzero weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which satisfies (14), (16), (17) with [kappa] = [infinity], (19) and the equality [[summation.sup.[infinity].sub.i=1][[absolute value of [[lambda].su.i,1]].sup.2] [[integral].sup.[infinity].sub.0] [s.sup.n]d [[mu].sub.i](s) = [infinity. Setting [[lambda].sub.i,1] = [square root of [[alpha].sub.i][q.sub.i]] for i [member of] N, we reduce our problem to find a

sequence [{[[alpha].sub.i]}.sup.[infinity].sub.i=1] [subset or equal to] (0, [infinity]) such that

[[infinity].summation over i=1] [[alpha].sub.i][q.sup.l.sub.i] < [infinity], l [member of] Z, l [less than or equal to] n, [[infinity].summation over i=1] [alpha].sub.i][q.sup.n+1.sub.i] = [infinity]. (21)

Indeed, if [{[[alpha].sub.i]}.sup.[infinity].sub.i=1] is such a sequence, then multiplying its terms by an appropriate positive constant, we may assume that [{[[alpha].sub.i]}.sup.[infinity].sub.i=1] satisfies (21) and 16). Next we define the weights {[[lambda].sub.-j] : j [member of] [Z.sub.+]} recursivelyso as to satisfy (17) with [kappa] = [infinity], and finally we set [[lambda].sub.i,j] = [square root of [q.sub.i]] for all i, j [member of] N such that j [greater than or equal to] 2. The so constructed weights [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] meet our requirements.

The following lemma turns out to be helpful when solving the reduced problem.

Lemma 7. If [[a.sub.i,j].sup.[infinity].sub.i, j=1] is an infinite matrix with entries [a.sub.i, j] [member of] [R.sub.+], then there exists a sequence [{[[alpha].sub.i]}.sup.[infinity].sub.i=1] [subset or equal to] (0, [infinity]) such that

[[infinity].summation over i=1] [[alpha].sub.1][[alpha].sub.i,j] < [infinity], j [member of] N. (22) Proof. First observe that, for every i [member of] N, there exists [[alpha].sub.i]] [member of] (0, [infinity]) such that [[alpha].sub.i][summation.sup.i.sub.k-1] [a.sub.i, k] [less than or equal to] [2.sup.-1]. Hence, [summation.sup.[infinity].sub.i=j] [[alpha].sub.i][[alpha].sub.i,j] [less than or equal to] 1 for every j [member of] N.

Since sup{[q.sub.i] : i [member of] N} = [infinity], there exists a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the sequence {[q.sub.i]}.sup.[infinity]sub.i=1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By Lemma 7, there exists {[[alpha].sub.i]}.sub.i[member of]N/[OMEGA]] [subset or equal to] (0, [infinity]) such that

[summation over i[member of]N/[OMEGA] [[alpha].sub.i][q.sup.l.sub.i] < [infinity], l [member of] Z, l [less than or equal to] n. (23) Define the system {[[alpha].sub.i]}.sub.i[member of][OMEGA]] [subset or equal to] (0, [infinity]) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get

TOTO

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Combining (23) and (25), we get (21), which solves the reduced problem and consequently gives the required example.

Remark 8. It is worth mentioning that if [kappa] = [infinity], then any weighted shift [S.sub.[lambda]] on [T.sub.[infinity],[infinity]] with nonzero weights is unitarily equivalent to an injective composition operator in an [L.sup.2]-space over a [sigma]-finite measure space (cf. [13, Lemma 4.3.1]). This fact combined with Example 1 shows that, for every n [member of] N, there exists a subnormal composition operator C in an [L.sup.2]-space over a [sigma]-finite measure space such that [C.sup.n] is densely defined and [C.sup.n+1] is not.

http://dx.doi.org/10.1155/2014/791817

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of the first author was supported by the NCN (National Science Center) Grant DEC-2011/01/D/ST1/05805. The research of the third and the fourth authors was supported by the MNiSzW (Ministry of Science and Higher Education) Grant NN201 546438 (2010-2013).

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Piotr Budzynski, (1) Piotr Dymek, (1,2) Zenon Jan Jablonski, (2) and Jan Stochel (2)

(1) Katedra Zastosowan Matematyki, Uniwersytet Rolniczy w Krakowie, Ulica Balicka 253c, 30-198 Krakow, Poland

(2) Instytut Matematyki, Uniwersytet Jagiellonski, Ulica Lojasiewicza 6, 30-348 Krakow, Poland

Correspondence should be addressed to Piotr Budzynski; piotr.budzynski@ur.krakow.pl

Received 15 October 2013; Accepted 3 December 2013; Published 19 February 2014

Academic Editor: Henryk Hudzik
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