# On some operator equations in the space of analytic functions and related questions/Monedest operaatorvorranditest analuutiliste funktsioonide ruumis.

1. INTRODUCTION AND BACKGROUND

Let B (E) be an algebra of all continuous linear operators acting on the topological vector space E. The operator equation

AX = XB (1)

naturally arises in numerous issues of spectral theory of operators, representation theory, stability theory (Lyapunov's equation), etc. For example, if the set of solutions of Eq. (1) contains a boundedly invertible operator [X.sub.0], then A and B are similar, B = [X.sup.-1.sub.0] A[X.sub.0], and hence have many common spectral properties. In general case, it is of interest to describe the set of all solutions of Eq. (1).

If B = [lambda]A, [lambda] [member of] C, then following , one refers to [lambda] as an extended eigenvalue of A, and each bounded solution X of the equation

AX = [lambda]XA,

i.e., Eq. (1) with B = [lambda]A, is called an extended eigenvector of A.

In this paper we investigate the so-called extended eigenvalues and extended eigenvectors and cyclicity problems for some convolution operators acting on the space of analytic functions defined on the starlike domain D of the complex plane. Our investigation is motivated by the results of Nagnibida's paper . By using the Duhamel product method (see ), we also give a lower estimate for the inner derivation operator [[DELTA].sub.A] defined in the Banach algebra B ([C.sup.(n).sub.A] (D)) by [[DELTA].sub.A] (X) := AX - XA.

The integration operator V on [L.sup.p] [0,1] (1 [less than or equal to] p < [infinity]) is defined by Vf (x) = [[integral].sup.x.sub.0] f (t) dt. The set of intertwining operators for the pair {[V.sup.[beta]], [lambda][V.sup.[beta]]} with [beta] > 0 and [lambda] [member of] C was studied by Malamud in [3,9,10]. Namely, he showed that there exists a nonzero intertwining operator for the pair {[V.sup.[beta]], [lambda][V.sup.[beta]]} only if [lambda] > 0. Furthermore, the paper  provides a description of the set [{[V.sup.[beta]}'.sub.[lambda]] of all intertwining operators for the pair {[V.sup.[beta]], [lambda][V.sup.[beta]]} for [lambda] > 0. For [beta] = 1, the latter result was reproved by another method by Biswas, Lambert, and Petrovic , and Karaev . For more details, see [1,2,4,5,9,10].

Let [alpha] be a fixed complex number, let D be a simply connected region in the complex plane C that is starlike with respect to the point z = [alpha] (i.e., [lambda]z + (1 - [lambda]) [alpha] [member of] D), and let A (D) be the space of all single-valued and analytic functions in D that have a topology of uniform convergence on compact subsets. It is well known that A (D) is a Frechet space. By [J.sub.[alpha]] we shall denote the integration operator in the space A(D) defined by the formula

([J.sub.[alpha]]f)(z) = [[integral].sup.z.sub.[alpha]] f (t) dt ([for all]f [member of] A (D)),

where the integration is performed over straight-line segments connecting the points [alpha] and z(z [member of] A(D)).

Recall that for f, g [member of] A (D) their [alpha]-Duhamel product is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where the integrals are taken over the segment joining the points [alpha] and z (z [member of] A (D)). It is easy to see that the [alpha]-Duhamel product satisfies all the axioms of multiplication, A (D) is an algebra with respect to [??] as well, and the function f (z) [equivalent to] 1 is the unit element of the algebra (A (D), [??]]. The operator [D.sub.f], [D.sup.[alpha].sub.f] g := f [??] g, is called the [alpha]-Duhamel operator on A(D).

2. EXTENDED EIGENVALUES AND EXTENDED EIGENVECTORS FOR SOME CONVOLUTION OPERATORS

Let D [subset] C be a starlike region with respect to the origin. For any fixed nonzero function f [member of] A (D), let [K.sub.f] be the usual convolution operator acting on the space A (D) by the formula

([K.sub.f]g) (z) = (f x g) (z) := [[integral].sup.z.sub.0] f (z - t) g (t) dt.

It follows from the classical Titchmarsh convolution theorem and uniqueness theorem for analytic functions that ker ([K.sub.f]) = {0}. This means that 0 is not an extended eigenvalue of the operator [K.sub.f], and therefore ext ([K.sub.f]) [subset] C\{0} (here ext ([K.sub.f]) denotes the set of all extended eigenvalues of the operator [K.sub.f]).

The integration operator J on A (D) is defined by J f (z) = [[integral].sup.z.sub.0] of (t) dt. Let [f.sup.[??]k] denote the [??]-product (which is clearly [??], that is the usual Duhamel product) of f with itself k times for k [greater than or equal to] 0, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [f.sup.[??]0] (z) [equivalent to] 1. If f is a function in A (D) such that [{[(Jf).sup.[??]n]}.sub.N[greater than or equal to]0] is a complete system in A (D), we will denote by [[LAMBDA].sub.f] the set of all [lambda] [member of] C\{0} for which the diagonal operator

[D.sub.{[lambda]}] [(Jf).sup.[??]n] = [[lambda].sup.n] [(Jf).sup.[??]n], n [greater than or equal to] 0,

is continuous in A(D).

The following theorem gives necessary and sufficient conditions under which the set [[LAMBDA].sub.f] lies in the set ext ([K.sub.f]). Our result is apparently the first result in the "extended theory" for more general convolution operators, which is an extension of Karaev's result [7, Theorem 2, (ii)]. The related results for the integration operator are considered in [4,7].

Theorem 1. Let f [member of] A (D) be a nonzero function. Suppose that the system [{[(Jf).sup.[??]n]}.sub.n[greater than or equal to]0] is complete in A (D). Let A [member of] B (A (D)) be a nonzero operator and [lambda] [member of] [[LAMBDA].sub.f] be any number. Then

A[K.sub.f] = [lambda][K.sub.f]A

if and only if there exists [phi] [member of] A (D) such that A = [D.sub.[phi]]D{X}.

Proof. By using the usual Duhamel product [??], which is defined by

([f.sub.1] [??] [f.sub.2]) (z) := [d/dz] [[integral].sup.z.sub.0] [f.sub.1] (z - t) [f.sub.2] (t) dt,

we have that any function [f.sub.1] [member of] A (D) defines the continuous operator (Duhamel operator) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then we have

[K.sub.f]g = J [D.sub.f]g = z [??] (f [??] g)

= (z [??] f) [??] g = [D.sub.z[??]f]g = [D.sub.Jf]g

for all g [member of] A (D). Thus [K.sub.f] = A [(D).sub.Jf].

Now, let [lambda] [member of] [[LAMBDA].sub.f] be any number, and suppose that

[lambda][K.sub.f]A = A[K.sub.f].

Then, obviously

[[lambda].sup.n][K.sup.f.sub.n]Ag = A[K.sup.n.sub.f]g

for all g [member of] A (D) and n [greater than or equal to] 0. In particular, putting g = 1 in the last equality, we have

A[K.sup.n.sub.f]1 = [[lambda].sup.n][K.sub.n.sub.f]A1

for all n [greater than or equal to] 0. Since [K.sub.f] = [D.sub.Jf], clearly we have

[K.sup.n.sub.f]1 = [D.sup.n.sub.Jf] 1 = [(Jf).sup.[??]n] [??] 1 = [(Jf).sup.[??]n]

for all n [greater than or equal to] 0. This shows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all n [greater than or equal to] 0. Since [{[(Jf).sup.[??]n]}.sub.n[greater than or equal to]0] is a complete system of the space A (D) and [D.sub.{[lambda]}] is a continuous operator on A (D), it follows from the last equalities that

Ag = [D.sub.A1][D.sub.{[lambda]}]g

for all g [member of] A (D), which means that A = [D.sub.[phi]][D.sub.{[lambda]}], where [phi] = A1 [member of] A (D), as desired.

Conversely, let us show that if A has the form A = [D.sub.[phi]]D{X}, where p [member of] A (D), it satisfies the equation A[K.sub.f] = [lambda][K.sub.f]A. In fact, by considering the formula A[K.sub.f] = [D.sub.Jf], and commutativity of the product [??], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all n [greater than or equal to] 0. By considering completeness of the system [{[(Jf).sup.[??]n]}.sub.n[greater than or equal to]0] in A(D), from the last equalities we deduce that A[K.sub.f] = [lambda][K.sub.f]A. The theorem is proved.

3. CYCLIC VECTORS OF CONVOLUTION OPERATOR [K.sub.f,[alpha]]

Let D be a starlike region in the complex plane C with respect to z = [alpha]. Our next result describes all cyclic vectors of some convolution operators of the form

([K.sub.f,[alpha]]g)(z) := [[integral].sup.z.sub.[alpha]] f (z + [alpha] - t)g(t) dt.

Theorem 2. Let f [member of] A (D), and assume that [{[([J.sub.[alpha]]f).sup.[??]n]}.sub.n[greater than or equal to]0] is a complete system in A (D). If g [member of] A (D), then g is a cyclic vector for the convolution operator [K.sub.f,[alpha]] if and only if g ([alpha]) [not equal to] 0.

Proof. It follows from the definition of [alpha]-Duhamel product [??] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all h [member of] A (D), which means that [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]f. Then according to the condition of the theorem, we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so

[E.sub.g] = clos[D.sub.g,[alpha]]A (D).

Now, if g ([alpha]) [not equal to] 0, then by virtue of Nagnibida's result operator [D.sub.g,[alpha]] is invertible in A(D), which implies that

[D.sub.g]A (D)= A (D).

Hence [E.sub.g] = A (D), which shows that g is a cyclic vector for the convolution operator [K.sub.f,[alpha]].

Conversely, suppose that g [member of] A(D) is a cyclic vector for the operator [K.sub.f,[alpha]], that is Eg = A(D). If g ([alpha]) [not equal to] 0, it is easy to see from the equality [E.sub.g] = clos [D.sub.g,[alpha]] A (D) that [E.sub.g] [subset] {h [member of] A (D) : h ([alpha]) = 0}, which is impossible because [E.sub.g] = A (D). Consequently, g ([alpha]) [not equal to] 0, which proves the theorem.

Since [{[(z - [alpha]).sup.n]}.sub.n[greater than or equal to]0] is a complete system in A (D), the next corollary immediately follows from Theorem 2.

Corollary 1. Let [J.sub.[alpha]] be an integration operator defined on A (D) by ([J.sub.[alpha]]g) (z) = [[integral].sup.z.sub.[alpha]] g (t) dt. Then

Cyc ([J.sub.[alpha]]) = {g [member of] A (D): g ([alpha]) [not equal to] 0},

where Cyc ([J.sub.[alpha]]) denotes the set of all cyclic vectors of [J.sub.[alpha]].

For the related results see [6-8] and Tkachenko ; in  the analogous results are considered by Karaev for the Banach space [C.sup.(n).sub.A] (D).

4. ON THE NORM OF INNER DERIVATION OPERATOR

Let A be a fixed linear bounded operator acting on the Banach space [C.sup.(n).sub.A] (D), which is the space of all n-times continuously differentiable functions on [bar.D] that are holomorphic on the unit disc D. In , Karaev proved that [C.sup.(n).sub.A] (D) is an algebra with multiplication of the Duhamel product

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Thus, the Duhamel operator [D.sub.f] defined on [C.sup.(n).sub.A] (D) by [D.sub.f]g := f [??] g is bounded and [parallel][D.sub.f][parallel] [less than or equal to] [parallel]f[parallel]. On the other hand, it is clear from (3) that f = f [??] 1, and therefore [parallel]Df[parallel] = [parallel]f[parallel]. In this section, by using this formula we will estimate the norm of the inner derivation operator [[DELTA].sub.A] defined on the Banach algebra B [C.sup.(n).sub.A] (D)) by the formula

[[DELTA].sub.A] (X) := AX - XA.

Obviously

[parallel][[DELTA].sub.A][parallel] [less than or equal to] 2[parallel]A[parallel]. (4)

The following theorem gives some lower estimate for [parallel][[DELTA].sub.A][parallel] in terms of A.

Theorem 3. Let A [member of] B ([C.sup.(n).sub.A] (D)) be a fixed operator. Suppose that for every X [member of] B ([C.sup.(n).sub.A] (D)) there exists a nonzero function f := [f.sub.X] [member of] [C.sup.(n).sub.A] (D) such that

((AX - XA) f) (0) [not equal to] 0.

Then there exists a constant [C.sub.A] > 0 such that

[C.sub.A] [less than or equal to] [parallel][[DELTA].sub.A][parallel] [less than or equal to] 2[parallel]A[parallel].

Proof. According to (4), there remains only to prove the left inequality. Indeed, let us denote

(AX - XA) f (z) := g (z). (5)

Clearly, g = [g.sub.A,X]. Since g(0) [not equal to] 0, by the result of paper [8, Theorem 1], the Duhamel operator [D.sub.g] is invertible in [C.sup.(n).sub.A] (D). Therefore, there exists a unique G [member of] [C.sup.(n).sub.A] (D) such that G [??] g = g [??] G = 1. Hence, f [??] G [??] g = f. Thus, it follows from (5) that

[D.sub.F] (AX - XA) f = f, (6)

where F := f [??] G. Clearly, F = [F.sub.A,X]. The equality (6) shows that 1 [member of] [[sigma].sub.p] ([D.sub.F] (AX - XA)), that is, 1 is the eigenvalue of the operator [D.sub.F] [[DELTA].sub.A] (X). Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

here r (.) denotes the spectral radius of the operator. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By taking supremum over the operators X with [parallel]X[parallel] [less than or equal to] 1, we have from this inequality that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By denoting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have the desired result. The theorem is proved.

doi: 10.3176/proc.2013.2.01

ACKNOWLEDGEMENTS

This work was supported by Siileyman Demirel University with Project 1918-YL-09. We are grateful to the referees for their important remarks and suggestions.

REFERENCES

[1.] Biswas, A., Lambert, A., and Petrovic, S. Extended eigenvalues and the Volterra operator. Glasgow Math. J., 2002, 44, 521-534

[2.] Bourdon, P. S. and Shapiro, J. H. Intertwining relations and extended eigenvalues for analytic Toeplitz operators. Ill. J. Math., 2008, 52, 1007-1030.

[3.] Domanov, I. Yu. and Malamud, M. M. On the spectral analysis of direct sums of Riemann--Liouville operators in Sobolev spaces of vector functions. Int. Equat. Oper. Theory, 2009, 63, 181-215.

[4.] Gurdal, M. Description of extended eigenvalues and extended eigenvectors of integration operator on the Wiener algebra. Expo. Math., 2009, 27, 153-160.

[5.] Gurdal, M. On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra. Appl. Math. Lett., 2009, 22, 1727-1729.

[6.] Karaev, M. T. Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some convolution operators. Methods Functional Anal. Topology, 2005, 11, 48-59.

[7.] Karaev, M. T. On some applications of ordinary and extended Duhamel products. Siberian Math. J., 2005, 46, 431-442 (translated from Sibirsk. Mat. Zh., 2005, 46, 553-566).

[8.] Karaev, M. T. and Saltan, S. A Banach algebra structure for the Wiener algebra W(D) of the disc. Complex Variables Theory Appl., 2005, 50, 299-305.

[9.] Malamud, M. M. Similarity of Volterra operators and related problems in the theory of differential equations of fractional orders. Trans. Moscow Math. Soc., 1995, 56, 57-122 (translated from Trudy Moskov. Mat. Obshch., 1994, 55, 73-148).

[10.] Malamud, M. M. Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators. Oper. Theory Adv. Appl., 1998, 102, 143-167.

[11.] Nagnibida, N. I. Description of commutants of integration operator in analytic spaces. Siberian Math. J., 1981, 22, 748-752 (translated from Sibirsk. Mat. Zh., 1981, 22, 127-131).

[12.] Tkachenko, V. A. Invariant subspaces and unicellularity of operators of generalized integration in spaces of analytic functionals. Math. Notes, 1997, 22, 221-230.

[13.] Wigley, N. M. The Duhamel product of analytic functions. Duke Math. J., 1974, 41, 211-217.

Mehmet Gurdal * and Filiz Sohret

Department of Mathematics, Suleyman Demirel University, 32260 Isparta, Turkey; filiz_sohret@hotmail.com

Received 4 October 2011, accepted 19 January 2012, available online 7 May 2013

* Corresponding author, gurdalmehmet@sdu.edu.tr
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