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Non-Weakly Supercyclic Classes of Weighted Composition Operators on Banach Spaces of Analytic Functions.

1 Introduction and preliminary results

Let X be a Banach space over the field of complex numbers and dim X > 1. By B(X) we mean the set of all bounded linear operators on X. A set [GAMMA] [??] B(X) is hypercyclic (supercyclic) if there exists a vector x [member of] X such that O(x, [GAMMA]) = {Tx : T [member of] r} ( C.O(x, [GAMMA]) = [[lambda]Tx : T [member of] [GAMMA], [lambda] [member of] C}) is a dense subset of X. An operator T [member of] B(X) is called hypercyclic (supercyclic) if the semigroup [GAMMA] = {[T.sup.n] : n [member of] [N.sub.0]} is hypercyclic (supercyclic). Here [N.sub.0] = {0, 1, 2, 3, ...}. Similarly, by considering density in the weak topology instead of the norm topology, we can define weak hypercyclicity and weak supercyclicity. A set [GAMMA] [??] B(X) is weakly hypercyclic (weakly supercyclic) if there exists a vector x [member of] X such that O(x,[GAMMA]) = {Tx : T [member of] [GAMMA]} (C.O(x, [GAMMA]) = [[lambda]Tx : T [member of] [GAMMA], [lambda] [member of] C}) is a weakly dense subset of X. An operator T [member of] B(X) is called weakly hypercyclic (weakly supercyclic) if the semigroup [GAMMA] = {[T.sup.n] : n [member of] [N.sub.0]} is weakly hypercyclic (weakly supercyclic).

Let D denote the open unit disc in the complex plane. By a Banach space of analytic functions we mean a Banach space consisting of analytic functions that contains the constant functions and the identity function such that the linear functional of point evaluation at [lambda] defined by [e.sub.[lambda]] (f) = f ([lambda]) is bounded for every [lambda] [member of] D. By the Cauchy integral formula, it is not surprising that evaluation of the derivative at each point [lambda] of the disc, which we denote by [e'.sub.[lambda]], is a bounded linear functional. In the following, some examples of classical Banach spaces of analytic functions are presented. Note that the class of all analytic functions on a simply connected domain [OMEGA] of the complex plane C will be denoted by H([OMEGA]), endowed with the compact open topology.

1. The space of all bounded analytic functions on D, denoted by [H.sup.[infinity]] with the norm [[parallel]f[parallel].sub.[infinity]] = [sup.sub.z[member of]D] |f(z)|.

2. The Hardy space [H.sup.p], 1 [less than or equal to] p < [infinity], consists of those functions f in H(D) for which [mathematical expression not reproducible] where dm is the normalized arc length measure on T, where T = {z [member of] C : |z| = 1}.

3. The space [S.sup.p], 1 [less than or equal to] p < [infinity], consisting of all analytic functions f on D for which

[mathematical expression not reproducible]

4. The weighted Bergman space [A.sup.p.sub.[alpha]] (1 [less than or equal to] p < [infinity], [alpha] > -1) is defined as the space of all f in H(D) such that

[mathematical expression not reproducible]

where dA is the normalized Lebesgue area measure on D. The space [A.sup.p.sub.0] is called the Bergman space and is denoted by [L.sup.p.sub.a].

5. The weighted Dirichlet-type space [D.sup.p.sub.[alpha]] (1 [less than or equal to] p < [infinity], -1 < [alpha] < [infinity]) is the space of f [member of] H(D) such that f' [member of] [A.sup.p.sub.[alpha]] equipped with the norm

[mathematical expression not reproducible]

6. The space [mathematical expression not reproducible] with the standard weights [v.sub.p](z) = [(1 - [|z|.sup.2]).sup.p] consisting of all analytic functions fon D such that

Non-Weakly Supercyclic Classes of Weighted Composition Operators

[mathematical expression not reproducible]

7. The analytic Besov space [B.sup.p], 1 < p < [infinity], is defined as the set of all analytic functions on the disc such that

[mathematical expression not reproducible]

8. The disc algebra A(D), is the Banach space of functions that are continuous on the closed unit disc and analytic on the open unit disc, with the supremum norm.

9. The analytic Lipschitz space [Lip.sub.[alpha]](D), (0 < [alpha] [less than or equal to] 1), is the set

{f analytic in D : |f(z) - f(w)| = O{[|z - w|.sup.[alpha]]) for all z, w [member of] [bar.D]}, with the norm

[[parallel]f[parallel].sub.[alpha]] = |f(0)| + sup{[|f(z) - f(w)|/[|z - w|.sup.[alpha]]] : z [not equal to] w [member of] [partial derivative]D}.

10. Let H be a Hilbert space whose vectors are functions analytic on D and the monomials 1, z, [z.sup.2],... constitute a complete orthogonal set of non-zero vectors in H. Writing [beta](j) = [parallel][z.sup.j][parallel], j [greater than or equal to] 1 with the normalization [beta](0) = 1, the inner product on H given by

[mathematical expression not reproducible]

The space H is called the weighted Hardy space with weight [beta] = [([beta](n)).sub.n[greater than or equal to]0] and will be denoted by [H.sup.2]([beta], D). See Pages 14 and 16 of [8].

Let [phi] be an automorphism of the disc. Recall that [phi] is elliptic if it has one fixed point in the disc and the other in the complement of the closed disc. Moreover, any elliptic automorphism [phi], with fixed point a [member of] D is conformally equivalent to [lambda]z where [lambda] = [phi]'(a) [8, Page 59]. For c [member of] D,let [[phi].sub.c](z) = (c - z)/(1 - cz), (z [member of] D). Throughout this paper, Aut(D) denotes the set of all automorphisms of D. Recall that a multiplier of a Banach space of analytic functions X is an analytic function w on D such that wX [??] X. The set of all multipliers of X is denoted by M(X). If w is a multiplier, then the multiplication operator [M.sub.w], defined by [M.sub.w]f = wf, is bounded on X. In what follows, suppose that w [member of] M(X) and [phi] is an analytic self map of D such that (f [??] [phi])(z) = f ([phi](z)) is in X for every f [member of] X. An application of the closed graph theorem shows that the weighted composition operator [C.sub.w,[phi]] defined by [C.sub.w,[phi]](f)(z) = [M.sub.w][C.sub.[phi]](f)(z) = w(z)f([phi](z)) is bounded. The map [phi] is called the composition map and w is called the weight. Throughout this paper, each self map of D is analytic.

In 1964, Forelli [11] showed that every isometry on [H.sup.p] for 1 < p < [infinity] and p [not equal to] 2 is a weighted composition operator. Recently, there has been a great interest in studying composition and weighted composition operators on the unit disc, polydisc, or the unit ball; see, for example, the monographs [8, 30], and the papers [4, 14].

Rolewicz [27] has shown that any scalar multiple [lambda]B of the unilateral backward shift B is hypercyclic on [l.sub.p] (1 [less than or equal to] p < [infinity]) whenever |[lambda]| > 1. On the other hand, in Kitai's dissertation [20], the linear dynamics, as a branch of functional analysis, was born. Recently, hypercyclic and supercyclic operators have been the focus of much work in linear dynamics. The reader can see [2] to get more information about hypercyclic and supercyclic operators. Sufficient conditions under which an operator is not weakly supercyclic are given by Montes-Rodriguez and Shkarin in [23]. Furthermore, Shkarin has studied non-sequential weak su-percyclicity and hypercyclicity of operators on a Banach space in [29]. Recently, Hedayatian and Faghih-Ahmadi have shown that every operator in the commutant of a cyclic convolution operator on the Hardy space [H.sup.p], p [greater than or equal to] 1 is not weakly supercyclic [16]. Moreover, several authors have studied the dynamics of a general weighted composition operator [C.sub.w,[phi]]. Recently, the hypercyclicity and supercyclicity of composition operators on H([OMEGA]) have been investigated with respect to the compact-open topology [5]. Also, the study of hypercyclicity and supercyclicity of composition operators on H(D) with respect to the weak topology and to its corresponding compact-open topology, have been explored in [19, 31]. Rezaei [26] studies composition operators that are chaotic in the sense of Devaney.

Recall that isometries cannot be supercyclic on Banach spaces [1, 22]. On the other hand, they can be weakly supercyclic. For example, surjective linear isometries can be weakly supercyclic on [l.sup.p](Z), (p > 2) [2, Page 253], and a unitary Hilbert space operator also can be weakly supercyclic [3]. The set of all unitary operators on a Hilbert space is supercyclic [10, Page 183] and so is weakly super-cyclic. Also, there are examples of supercyclic groups of isometries on Banach spaces. The group of all isometries on [L.sup.u]([mu]) (1 [less than or equal to] p < [infinity]) where [mu] is a homogeneous measure is supercyclic [13]. Moreover, for 1 [less than or equal to] p < [infinity], the group of all isometries on the Banach space [L.sup.p](X, [mu]) is supercyclic, where X is the disjoint union of an uncountable family of copies of the interval [0, 1] and [mu] is a certain measure on X [15]. It has been shown that the semigroup of linear isometries on the space [S.sup.p] (p > 1), the group of all surjective linear isometries on the Hardy space [H.sup.p] and the Bergman space [L.sup.p.sub.a] (1 < p < [infinity], p [not equal to] 2) are not supercyclic [24]. In Section 2, we give sufficient conditions for non-weak supercyclicity of vectors in an infinite dimensional Banach space. In particular, we observe that the semigroup of linear isometries on the space [S.sup.p] (p > 1) is not weakly super-cyclic. In Section 3, we study the weak supercyclicity behavior of certain classes of weighted composition operators on some spaces of analytic functions on the unit disc D. Also, we show that the class : [C.sub.[phi]]:[C.sub.[phi]] is an isometry} is not weakly super-cyclic on the Hardy space [H.sup.p], the weighted Bergman space [A.sup.p.sub.[alpha]] (1 [less than or equal to] p < [infinity]), the analytic Besov space [B.sup.p] (2 < p < [infinity]), and the space [mathematical expression not reproducible]. Moreover, we observe that every composition operator on some Banach spaces of analytic functions such as the disc algebra or the analytic Lipschitz space is not weakly supercyclic. Furthermore, sufficient conditions are given for non-weak super-cyclicity of a weighted composition operator.

2 Infinite dimensional reflexive Banach spaces

The angle-criterion for supercyclicity states that for a T [member of] B(X) and x [member of] X, if there is a non-zero x* [member of] X* such that

[??]|[x*([T.sup.n]x)|/[parallel][T.sup.n]x[parallel][parallel]x*[parallel]] [not equal to] 1

then x is not a supercyclic vector for T. Using this criterion it is shown that the classical Volterra operator and the composition operators associated with parabolic non-automorphisms of the unit disc D are not supercyclic, (see for example, Section 9.1 of [2]) . In this section, we give a non-weak supercyclicity criterion for vectors in an infinite dimensional reflexive Banach space which is, to some extent, similar to the angle-criterion but has a quite different method of proof. By applying this criterion we show that various operators are not weakly supercyclic.

Given a directed set I, a Banach space X, [x.sub.i] [member of] X (i [member of] I) and x [member of] X, we use the expression [x.sub.i] [??] x to denote that the net [([x.sub.i]).sub.i[member of]I] converges to x with respect to the weak topology of X. Also, the weak operator topology (WOT) on B(X) is the one in which a net ([T.sub.[alpha]]) converges to T if and only if [T.sub.[alpha]](x) [right arrow] T(x) weakly for all x [member of] X.

Theorem 1. Let X be an infinite dimensional reflexive Banach space, [GAMMA] be a subset of B(X) and x be a non-zero vector in X. If there exists a linear functional x* [member of] X * such that

c = inf{[|x*(Tx)|/[parallel]T[parallel]] : T [member of] [GAMMA]} > 0, then x is not a Weakly supercyclic vector for [GAMMA]. In addition, the set of all Weakly supercyclic vectors for [GAMMA] is not norm dense in X.

Proof. Since the weak supercyclicity of {[T/[parallel]T[parallel]] : T [member of] [GAMMA]} is equivalent to the weak supercyclicity of r, one can assume that [parallel]T[parallel] = 1 for all T [member of] r. Suppose, on the contrary, that x is a weakly supercyclic vector for r. Therefore, for any y [member of] X there are two nets [([[alpha].sub.i]).sub.i] in C and [([T.sub.i]).sub.i] in [GAMMA] such that [[alpha].sub.i][T.sub.i](x) [??] y. Thus, there exists j so that

|x*([[alpha].sub.i][T.sub.i](x)) - x*(y)| < 1 for all i > j,

which in turn implies that

|[[alpha].sub.i]| [less than or equal to] [c.sup.-1](|x*(y)| + 1) for all i > j.

By passing to a subnet if necessary, we assume that [lim.sub.i] [[alpha].sub.i] = [alpha] exists. Since the unit ball of B(X) is WOT-compact, [([T.sub.i]).sub.i] has a WOT-convergent subnet [mathematical expression not reproducible] with limit T [member of] [[GAMMA].sup.WOT]. So we observe that [mathematical expression not reproducible] and hence [alpha]Tx = y, which implies that

X= {[alpha]Tx:[alpha] [member of] C, T [member of] [[bar.[GAMMA]].sup.WOT]}

Since the weak closure of the unit sphere of X is ball(X), there exists a net [([x.sub.[beta]]).sub.[beta]] of unit vectors weakly converging to zero. On the other hand, for each [beta] there is an operator [S.sub.[beta]] [member of] [[bar.[GAMMA]].sup.WOT] and [[lambda].sub.[beta]] [member of] C such that [[lambda].sub.[beta]][S.sub.[beta]](x) = [x.sub.[beta]]. Thus

c|[[lambda].sub.[beta]]| [less than or equal to] |[[lambda].sub.[beta]]||x*([S.sub.[beta]](x))| = |x*([[lambda].sub.[beta]])|

and

1 = [parallel][[lambda].sub.[beta]][S.sub.[beta]](x) [parallel][less than or equal to] |[[lambda].sub.[beta]]| [parallel] x[parallel]

show that

[1/[parallel]x[parallel]] [less than or equal to] |[[lambda].sub.[beta]]| [less than or equal to] [c.sup.-1][parallel]x*[parallel]].

It follows that there exist subnets [mathematical expression not reproducible] and [mathematical expression not reproducible] such that [mathematical expression not reproducible], for some non-zero constant [lambda], and [mathematical expression not reproducible]. Hence

[mathematical expression not reproducible]

which is absurd.

For the next part put V = {y [member of] X : [parallel]y - x[parallel] < c/2[parallel]x*[parallel]}. Therefore, for y [member of] V and T [member of] [GAMMA]

|x* (Ty)| [greater than or equal to] |x* (Tx)| - |x* (Ty - Tx)|

[greater than or equal to] c[parallel]T[parallel]/2.

Thus, y is not a weakly supercyclic vector for [GAMMA].

Corollary 1. Let X be an infinite dimensional reflexive Banach space, [GAMMA] be a bounded subset of B(X) and x be a non-zero vector in X. If there exists a linear functional x* [member of] X* such that

c = inf {|x*(Tx)| : T [member of] [GAMMA]} > 0, then x is not a weakly supercyclic vector for [GAMMA]. Moreover, the set ofall weakly supercyclic vectors for [GAMMA] is not norm dense in X.

In the preceding theorem let x be the constant function 1 and x* be the evaluation functional at a. Then we have the following corollaries.

Corollary 2. Let X be an infinite dimensional reflexive Banach space of analytic functions on the open unit disc D and [GAMMA] be a class ofweighted composition operators on X which is bounded. Suppose that there is a [member of] D and [epsilon] > 0 such that |w(a)| > [epsilon] for all weighted maps w such that [C.sub.w,[phi]] [member of] [GAMMA], for some composition map [phi]. Then the set of weakly supercyclic vectors for [GAMMA] is not norm dense in X.

Corollary 3. The set of weakly supercyclic vectors for any bounded set of composition operators on an infinite dimensional reflexive Banach space of analytic functions on D is not norm dense in X.

Let [phi] be a self map on D. For a positive integer n, the nth iterate of [phi] is denoted by [[phi].sub.n] and [[phi].sub.0] is the identity function. Also, for a weighted composition operator [C.sub.w,[phi]] on a Banach space of analytic function X, we have

[C.sup.n.sub.w,[phi]](f) = [[product].sup.n-1.sub.j=0]w [??] [q.sub.j]. (f [??] [[phi].sub.n]),

for all f [member of] X and n [greater than or equal to] 1. Also, recall that an operator T is a power bounded operator, if there exists a positive number M such that [parallel][T.sup.n][parallel] [less than or equal to] M for all n [greater than or equal to] 0. Taking [GAMMA] = {[C.sup.n.sub.w,[phi]] : n [greater than or equal to] 0} in Corollary 1, with x* the evaluation functional at a we obtain the following result.

Corollary 4. Let [C.sub.w,[phi]] be a weighted composition operator on an infinite dimensional reflexive Banach space of analytic functions. Suppose that [phi] has a fixed point a [member of] D such that |w(a)| [greater than or equal to] 1.

1. If [C.sub.w,[phi]] is power bounded then it is not weakly supercyclic.

2. If [C.sub.w,[phi]] is weakly supercyclic then [parallel][C.sub.w,[phi]][parallel] > 1.

Proposition 1. Let X be an infinite dimensional reflexive Banach space and T be a power bounded operator on X which has a non-zero fixed point. Then T is not weakly super-cyclic.

Proof. Assume on the contrary that T is weakly supercyclic. By [28, Proposition 2.1 ] the set of weakly supercyclic vectors for T is norm dense in X. Let x be a non-zero fixed point of T. By the Hahn-Banach Theorem there is x* [member of] X* such that x* (x) [not equal to] 0. Hence | x* ([T.sup.n]x)| = |x* (x)| > 0 for all n > 0 and by Theorem 1 we have a contradiction.

Since [C.sub.[phi]]1 = 1 we have the following result.

Corollary 5. Suppose that X is an infinite dimensional reflexive Banach space of analytic functions. If [C.sub.[phi]] is weakly supercyclic on X then [parallel][C.sub.[phi]][parallel] > 1.

Corollary 6. Let X be the Hardy space [H.sup.p] or the Bergman space [L.sup.p.sub.a], (1 < p < [infinity]). If [phi], not the identity and not an elliptic automorphism, has a fixed point a [member of] D, then [C.sub.[phi]] is not weakly supercyclic.

Proof. It is known that [lim.sub.n[right arrow][infinity]] [[phi].sub.n] (0) = a (see page 59 of [8]). Moreover, by Corollary 3.7 of [8] and Theorem 10.3.2 of [32],

[parallel][C.sup.n.sub.[phi]][parallel] [less than or equal to] [([1 + |a|/1 - |a|]).sup.[beta]],

hence

[??][parallel][C.sup.n.sub.[phi]][parallel] [less than or equal to] [([1 + |a|/1 - |a|]).sup.[beta]],

where [beta] = [1/p] for the space [H.sup.p] and [beta] = [2/p] for the space [L.sup.p.sub.a]. Thus, the result follows from Proposition 1 or Corollary 4.

Definition 1. Let X be a Banach space of analytic functions on D. A net [([T.sub.i]).sub.i] in B(X), is said to converge pointwise evaluation to T [member of] B(X) if [lim.sub.i[right arrow][infinity]] ([T.sub.i]f)(z) = (Tf )(z) for all f [member of] X and all z [member of] D; this property is denoted by [T.sub.i] [??] T.

Theorem 2. Let X be an infinite dimensional Banach space of analytic functions on D and f [member of] X. Suppose that [GAMMA] is a subset of B(X) such that each net [([T.sub.i]).sub.i] in [GAMMA] has a subnet which converges pointwise evaluation to some T [member of] [GAMMA] and moreover, [GAMMA] is bounded away from 0 and [infinity], i.e., there exist two positive constants [c.sub.1] and [c.sub.2] such that [c.sub.1] [less than or equal to] [parallel]T[parallel] [less than or equal to] [c.sub.2] for all T [member of] [GAMMA]. Ifthere exists [lambda] [member of] D such that

c= inf{[|(Tf)([lambda])|/[parallel]T[parallel]]:T [member of] [GAMMA]} > 0,

then f is not a weakly supercyclic vector for [GAMMA]. In addition, the set of weakly supercyclic vectors for [GAMMA] is not norm dense in X.

Proof. Suppose that on the contrary, f is a weakly supercyclic vector for r. Therefore, for any g [member of] X there are two nets [([[alpha].sub.i]).sub.i] in C and [([T.sub.i]).sub.i] in [GAMMA] such that

[[alpha].sub.i][T.sub.i](f) [??] g. Thus, there exists j so that

|([[alpha].sub.i][T.sub.i](f))([lambda]) - g([lambda])|< 1 for all i > j

which in turn implies that

|([[alpha].sub.i] [less than or equal to] [([c.sub.1]c).sup.-1] (|g([lambda])|< 1 for all i > j

By passing to a subnet if necessary, we assume that [lim.sub.i] [[alpha].sub.i] = a exists. By hypothesis there is a subnet [mathematical expression not reproducible] so that [mathematical expression not reproducible] for some T [member of] [GAMMA]. So we observe that [mathematical expression not reproducible] for all z [member of] D and hence [alpha]Tf = g, which implies that

X = {[alpha]Tf : [alpha] [member of] C, T [member of] [GAMMA]}.

Since the weak closure of the unit sphere of X is ball(X), there exists a net [([f.sub.[beta]]).sub.[beta]] of unit vectors weakly converging to zero. On the other hand, for each [beta] there is an operator [S.sub.[beta]] [member of] [GAMMA] and [[lambda].sub.[beta]] [member of] C such that [[lambda].sub.[beta]][S.sub.[beta]] (f) = [f.sub.[beta]]. Thus

c[c.sub.1]|[[lambda].sub.[beta]]| [less than or equal to] c |[[lambda].sub.[beta]][parallel]|[S.sub.[beta]][parallel] [less than or equal to] |[[lambda].sub.[beta]]|[S.sub.[beta]](f)([lambda]]| = |[e.sub.[lambda]]([f.sub.[beta]])|

which implies that

[1/[c.sub.2][parallel]f[parallel]] [less than or equal to] |[[lambda].sub.[beta]]| [less than or equal to] [[parallel][e.sub.[lambda]][parallel]/[c.sub.1]c]

It follows that there exist subnets [mathematical expression not reproducible] and [mathematical expression not reproducible] such that [mathematical expression not reproducible], for some non-zero constant [lambda]', and [mathematical expression not reproducible]. Hence

[mathematical expression not reproducible]

which is absurd. For the next part put V = {g [member of] X : [parallel]g - f[parallel] < c/2[parallel][e.sub.[lambda]][parallel]}.

Therefore, for g [member of] V and T [member of] [GAMMA]

|Tg([lambda])| [greater than or equal to] |Tf ([lambda])| -|Tg([lambda]) - Tf ([lambda])|

[greater than or equal to] c[parallel]T[parallel]/2.

Thus, g is not a weakly supercyclic vector for [GAMMA].

There exists a norm equivalent to the original norm of a Banach space X such that the group of all surjective linear isometries on X with the new norm consists only of unimodular scalars of the identity [18]. Thus, one can find a Banach space on which the group of surjective linear isometries is not supercyclic; even stronger is not weakly supercyclic. Recall that the big Bloch space B is the set of all analytic functions f on D such that f (0) = 0 and

[parallel]f[parallel] = sup{|f'(z)|(1 - |[z.sup.2]|) : z [member of] D} < [infinity].

Also, the little Bloch space B0, is the subspace of B spanned by the polynomials. The set of all linear isometries on the little Bloch space [B.sub.0] and the set of all surjective linear isometries on the big Bloch space B has the form [GAMMA] = {[lambda]([C.sub.[phi]] - [Z.sub.[phi]]) : [phi] is a rotation, |[lambda]| = 1} where [Z.sub.[phi]](f) = f ([phi](0)) [9]. Since [e.sub.0] is continuous, there is no weakly supercyclic vector for this spaces. Therefore, the semigroup of linear isometries on the little Bloch space [B.sub.0] and the group of surjective linear isometries on the big Bloch space B are not weakly supercyclic. On the other hand, the group of all unitary operators on a Hilbert space is supercyclic. Thus, it is natural to investigate whether the semigroup of linear isometries or the group of surjective linear isometries on a Banach space is weakly supercyclic or not.

Proposition 2. The semigroup oflinear isometries on the space [S.sup.p], p > 1, is not weakly supercyclic.

Proof. The set of all isometries on [S.sup.p], p > 1 is of the form

[GAMMA] = {[beta][C.sub.[phi]] : [phi] is a rotation and |[beta]| = 1}

[25]. Suppose on the contrary that f is a weakly supercyclic vector of [GAMMA]. Since point evaluations are continuous, {[alpha]f (0) : [alpha] [member of] C} is dense in C. Thus f (0) [not equal to] 0. Let [mathematical expression not reproducible] be an arbitrary net in r, where |[[beta].sub.i]| = 1 and [[phi].sub.i] (z) = [[lambda].sub.i]z for some [[lambda].sub.i] with |[[lambda].sub.i] | = 1 and each z [member of] D. There exist two subnets [mathematical expression not reproducible] and [mathematical expression not reproducible] such that [mathematical expression not reproducible] and [mathematical expression not reproducible] for some numbers |[beta]| = 1 and |[lambda]| = 1 . Hence [mathematical expression not reproducible] where T = [beta][C.sub.[phi]] and [phi](z) = [lambda]z for each z [member of] D. Put [lambda] = 0 in Theorem 2, thus we get a contradiction.

3 Classes of weighted composition operators

In this section, we give necessary conditions for weak supercyclicity of certain classes of weighted composition operators on a Banach space of analytic functions.

Theorem 3. Let X be a Banach space of analytic functions on D and [GAMMA] [??] B(X) be a class of weighted composition operators on X such that for two points; a, b [member of] D, [GAMMA] [??] {[C.sub.w,[phi]] : [phi](a) = b, w(a) [not equal to] 0}. If [GAMMA] is weakly supercyclic, then the sets

A = {[w(c)/w(a)] : [there exists] [phi] such that [C.sub.w,[phi]] [member of] [GAMMA]},

and

B = {[w'(a)/w(a)] : [there exists] [phi] such that [C.sub.w,[phi]] [member of] [GAMMA]},

are unbounded for every c [member of] D, c [not equal to] a.

Proof. Suppose that f is a weakly supercyclic vector of [GAMMA]. Since point evaluations are continuous, {[alpha]w(a)f (b) : [alpha] [member of] C, [C.sub.w,[phi]] [member of] [GAMMA]} is dense in C. Thus f (b) [not equal to] 0. Let g(z) = z - a, [epsilon] > 0 and c [member of] D, c [not equal to] a. Put

[U.sub.g] = {h [member of] X : |[e.sub.a](h - g)| < [epsilon],|[e.sub.c](h - g)| < [epsilon]}

a weak neighborhood of g. There exist [[alpha].sub.0] [member of] C and [mathematical expression not reproducible] such that

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Since, for any self map [phi] of D, [[phi].sub.b][phi][[phi].sub.a](0) = 0 the Schwarz lemma implies that

|[[phi].sub.b][phi][[phi].sub.a](z)| [less than or equal to] |z| (z [member of] D);

thus,

|[[phi].sub.b][phi][[phi].sub.a](z)| [less than or equal to] |z| (z [member of] D).

Therefore, there is a constant M such that

[??]{|f ([phi](c))| : [phi] is a self map of D with [phi](a) = b} [less than or equal to] M, since f [??] [phi] = (f [??] [phi]b) [??] ([[phi].sub.b] [??] [phi]). Hence

[mathematical expression not reproducible]

Now if the set {[w(c)/w(a)]: [there exists] [phi] such that [C.sub.w,[phi]] [member of] [GAMMA]] is bounded then c = a which is a contradiction. On the other hand, for g(z) = z - a one can find a net [mathematical expression not reproducible] in C.[GAMMA](f) such that

[mathematical expression not reproducible]

Therefore,

[[alpha].sub.i][w.sub.i](a)f(b) [right arrow] 0.

Moreover, since [e'.sub.a] is continuous, we have

[[alpha].sub.i] ([w'.sub.i](a)f (b) + [w.sub.i] (a) [[phi]'.sub.i] (a)f'(b)) [right arrow] 1.

But an application of the Cauchy integral formula shows that there is a constant M so that |[[phi]'.sub.i] (a)| [less than or equal to] M, for all i; hence

[[alpha].sub.i][w'.sub.i](a) f (b) [right arrow] 1.

Thus, [w'.sub.i](a)/[w.sub.i](a) [right arrow] [infinity], which implies that the set {[w'(a)/w(a)]: [there exists] [phi] such that [C.sub.w,[phi]] [member of] [GAMMA]} is unbounded.

As a consequence of the preceding theorem, suppose that [C.sub.w,[phi]] is weakly supercyclic on a Banach space of analytic functions X. If a [member of] D is a fixed point of [phi] and w(a) [not equal to] 0 then the sequence [([[[product].sup.n-1.sub.j=0]w[??][[phi].sub.j](c)/[w.sup.n](a)]).sub.n] is unbounded for every c [member of] D - {a}. This improves Theorem 2.2 of [19].

Also, Corollary 6 can be extended as follows.

Corollary 7. Let X be a Banach space of analytic functions on D and a, b [member of] D. Then the semigroup [GAMMA] = {[C.sub.[phi]]: [phi](a) = b} is not weakly supercyclic. In particular, if [phi] has a fixed point then [C.sub.[phi]] is not weakly supercyclic.

In addition, the semigroup [GAMMA] = : {[C.sub.[phi]] : [C.sub.[phi]] is an isometry} is not weakly supercyclic on the Hardy space [H.sup.p], the weighted Bergman space [A.sup.p.sub.[alpha]] (1 [less than or equal to] p < [infinity]), the analytic Besov space [B.sup.p] (2 < p < [infinity]) and the space [mathematical expression not reproducible]. Moreover, the semigroup [GAMMA] = : {[C.sub.[phi]] : [phi] [member of] Aut(D) and [C.sub.[phi]] is an isometry} is not weakly supercyclic on the weighted Dirichlet-type space [D.sup.p.sub.[alpha]] (1 [less than or equal to] p < [infinity], -1 < [alpha] < [infinity]). For all of these spaces [GAMMA] = {[C.sub.[phi]]: [phi] is a rotation}. The relevant references are, respectively, [21, Theorems 1.3 and 1.4], [6, Corollary 12] and [12, Corollary 2.3]. Thus, the result follows from the previous corollary.

Proposition 3. Let X be a Banach space of analytic functions on D and T = [C.sub.w,[phi]] for which [phi] is not the identity and not an elliptic automorphism, has a fixed point a [member of] D, such that w(a) [not equal to] 0. Then T is not weakly supercyclic.

Proof. Since

[C.sup.n.sub.w,[phi]](f) = [[product].sup.n-1.sub.j=0]w [??] [[phi].sub.j]. (f [??] [[phi].sub.n]), for all f [member of] X and n [greater than or equal to] 1, [W.sub.n] = [[product].sup.n-1.sub.j=0]w [??] [[phi].sub.j] is the weight of [T.sup.n] for n [greater than or equal to] 1. Moreover,

[W'.sub.n](a) = w'(a)[w.sup.n-1] (a)(1 + [n-1.summation over (j=1)][[phi]'.sup.j](a)),

and

[W.sub.n](a) = [w.sup.n](a);

thus,

[[W'.sub.n](a)/[W'.sub.n](a)] = [w'(a)/w(a)] (1 + [n-1.summation over (j=1)][[phi]'.sup.j](a)).

On the other hand, |[phi]'([alpha])| < 1 [8, Page 59]. Hence [[W'.sub.n](a)/[W'.sub.n](a)] is a bounded set and the result follows from Theorem 3.

Recall that the group of surjective linear isometries on the Hardy space [H.sup.p] or the Bergman space [L.sup.p.sub.a] (1 < p < [infinity], p [not equal to] 2) have the form of weighted composition operators; i.e., {[mu][([phi]').sup.[beta]][C.sub.[phi]] : |[mu]| = 1, [phi] [member of] Aut(D)} where [beta] = [1/p] or [beta] = [2/p], respectively ([9],[17]).

Corollary 8. Let X be a Banach space of analytic functions on D, a [member of] D and [beta] > 0. If [GAMMA] [??] {[mu][([phi]').sup.[beta]][C.sub.[phi]] : |[mu]| = 1, [phi] [member of] Aut(D), [phi](a) = a} then [GAMMA] is not weakly supercyclic.

Proof. Suppose that [mu][([phi]').sup.[beta]][C.sub.[phi]] [member of] [GAMMA] and put w = [mu][([phi]').sup.[beta]]. Observe that there are c [member of] D and [lambda] with |[lambda]| = 1 such that [phi] = [[lambda][phi].sub.c]. Thus

|w(a)| = |[([|c|.sup.2] - 1).sup.[beta]] [(1 - [bar.c]a).sup-2[beta]|

and

|w'(a)| = |2[beta][bar.c][([|c|.sup.2] - 1).sup.[beta]] [(1 - [bar.c]a).sup.-2[beta]-1]|.

Therefore,

sup{[|w'(a)|/|w(a)|] : [there exists] [phi] such that [C.sub.w,[phi]] [member of] [GAMMA]} < [infinity]

and the result follows from Theorem 3.

Corollary 9. Let X be the Hardy space [H.sup.p] or the Bergman space [L.sup.p.sub.a] (1 < p < [infinity], p [not equal to] 2) and T = [mu][([phi]').sup.[beta]][C.sub.[phi]] be a surjective linear isometry on X for some [phi] [member of] Aut(D) and |[mu]| = 1 where [beta] = [1/p] for the space [H.sup.p] and [beta] = [2/p] for the space [L.sup.p.suba.].If T is weakly supercyclic then [phi] is not an elliptic automorphism.

Note that the weighted composition operator [C.sub.w,[phi]] is unitary on [H.sup.2](D) if and only if [phi] is an automorphism of D and w = [mu][([phi]').sup.[1/2]] for some |[mu]| = 1 [7, Theorem 6]. Therefore, if [C.sub.w,[phi]] is weakly supercyclic then [phi] is not an elliptic automorphism.

Let Y be a Banach space of analytic functions on D such that every element in Y has a continuous extension on [bar.D] and for every [lambda] [member of] [partial derivative]D the linear functional of point evaluation at [lambda], [e.sub.[lambda]] is bounded. The disc algebra A(D), the analytic Lipschitz [Lip.sub.[alpha]](D), (0 < [alpha] [less than or equal to] 1), [S.sup.p], (p [greater than or equal to] 1) and [H.sup.2]([beta], D) when [SIGMA][beta][(n).sup.-2] < [infinity], are some examples of such spaces [8, Pages 28 and 177].

The proof of the following proposition is the same as the proof of Theorem 3. Also, we assume that the function [phi] is continuous on [bar.D] and note that w = [C.sub.w,[phi]] [member of] Y; therefore, w is continuous on [bar.D].

Proposition 4. Let [GAMMA] [??] B(Y) be a class of weighted composition operators on Y such that for two points a, b [member of] [bar.D], [GAMMA] [??] {[C.sub.w,[phi]] : [phi]([alpha]) = b, w{a) [not equal to] 0}. If [GAMMA] is weakly supercyclic, then the set {[w(c)/w(a)] : [there exists] [phi] such that [C.sub.w,[phi]] [GAMMA]} is unbounded for every c [member of] [bar.D], c [not equal to] a.

Corollary 10. Let a, b [member of] [bar.D] and [GAMMA] = {[C.sub.[phi]] : [phi]{a) = b}. Then [GAMMA] is not weakly super-cyclic. In particular, by the Brouwer's fixed-point theorem every composition operator on Y is not weakly supercyclic.

Acknowledgments. We thank the referee for his helpful comments that improved the presentation of this paper. This research was in part supported by a grant from Shiraz University Research Council.

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A. Moradi B. Khani Robati K. Hedayatian (*)

Department of Mathematics, College of Sciences, Shiraz University, Shiraz 7146713565, Iran

e-mails: amoradi@shirazu.ac.ir, bkhani@shirazu.ac.ir and hedayati@shirazu.ac.ir, khedayatian@gmail.com

(*) Corresponding author

Received by the editors in December 2015 - In revised form in October 2016.

Communicated by F. Bastin.

Key words and phrases : Weakly supercyclic, composition operators, semigroup, isometry, fixed point.
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