# Topological Transitivity of Shift Similar Operators on Nonseparable Hilbert Spaces

Authors: Andriy Zagorodnyuk; Zoriana Novosad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.


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## Body

## 1. Introduction

LetX be a Hausdorff locally convex space. A continuous operator T:X⟶X is called topologically transitive if for each pair U,V of nonempty open subsets of X, there is some n∈ℕ with TnU∩V≠∅. If the underlying space is a separable Baire space, the transitivity is equivalent to the hypercyclicity by Birkhoff’s transitivity theorem (see [1], p. 2). A continuous linear operator T:X⟶X acting on a separable Fréchet space X is called hypercyclic if there is a vector x∈X for which the orbit under T,(1)OrbT,x=x,Tx,T2x,⋯is dense in X. Every such vector x is called a hypercyclic vector of T.The study of topological transitivity was started from Birkhoff’s result [2] in 1929 for nonlinear continuous functions. This result is important in the theory of chaos (see e.g., [3]). For the space of all entire functions, Birkhoff [2] also proved that the translation operator Ta:fx↦fx+a,a≠0 is hypercyclic. MacLane in 1952 showed that the differentiation operator Tf=f′ is hypercyclic too (see [4]).A lot of work on hypercyclicity has been based on the well-known so-called hypercyclicity criterion (the Kitai-Gethner-Shapiro theorem). In [5], Bermúdez and Kalton observed that similar criterion holds for topologically transitive operators on Banach spaces.Theorem 1 (topologically transitive criterion).
LetT be a bounded linear operator on a complex Banach space X (not necessarily separable). Suppose that there exists a strictly increasing sequence nk of positive integers for which there are
(i)
A dense subsetX0⊂X such that Tnkx⟶0 for every x∈X0 as k⟶∞(ii)
A dense subsetY0⊂X and a sequence of mappings Sk:Y0⟶X such that Sky⟶0 for every y∈Y0 and Tnk∘Sky⟶y for every y∈Y0 as k⟶∞
Then,T is topologically transitive.More information on transitive and hypercyclic operators can be found in [1, 6]. The existence of an element with a dense orbit implies that X must be separable. On the other hand, the transitivity of T:X⟶X does not require that the space X is separable.Problem 2 (Bermúdez and Kalton [5]).
Is there any characterization of nonseparable Banach spaces which support a topologically transitive operator?
Note that any hypercyclic operatorT:X⟶X always has an invariant, norm dense, linear subspace in which every nonzero vector is a hypercyclic vector for T.Theorem 3 (see [7]).
LetX be a complete metric space which has no isolated points. Let T:X⟶X be topologically transitive operator and y∈X. There exists a dense Gδ subset D of X such that for each z∈D, there exists a T-invariant (separable) closed subspace Yz of X with y,z∈Yz such that the restriction of T to Yz,T:Yz⟶Yz, is hypercyclic.The hypercyclisity of composition operators and differentiation operators on some function spaces of finite and infinite variables was studied in [8–14]. According to a classical theorem of Rolewicz [15], the weighted backward shifts on ℓ2(2)x1,x2,⋯,xn,⋯↦λx2,λx3,⋯,λxn+1,⋯is hypercyclic if λ>1. This fact remains to be true if we replace ℓ2 into ℓp for 1≤p<∞. Bermúdez and Kalton in [5] showed that spaces like ℓ∞ and Lℓ2) do not support topologically transitive operators, where Lℓ2 is the space of all bounded operators on ℓ2.In [7] (Proposition 3.4), Manoussos proved that the backward weighted shift is topologically transitive on nonseparable Hilbert space. More detailed, let H be a Hilbert space (possibly nonseparable) and ℓ2H be the ℓ2-sum of infinitely many copies of H. That is, if x∈ℓ2H, then
(3)x=x0,x1,⋯,xn,⋯,xn∈H,x2=∑n=0∞xn2.Then, thebackward weighted shiftTwith weight sequenceωn is defined by the following:
(4)T:x0,x1,⋯,xn,⋯↦ω1x1,ω2x2,⋯,ωnxn,⋯.Unfortunately, Proposition 3.4 in [7] contains a slight inaccuracy which for us is essential. So, we propose the corrected version.Proposition 4.
LetH be a Hilbert space and T:ℓ2H⟶ℓ2H be a backward weighted shift with positive weight sequence ωn. The following are equivalent:
(i)
T is topologically transitive(ii)
There exists a nontrivialT-invariant (separable) closed subspace Y⊂ℓ2H on which the restriction of T to Y,T:Y⟶Y, is hypercyclic(iii)
The restrictionT:Y⟶Y to any T-invariant (separable) closed subspace Y⊂ℓ2H which contains nonzero vectors of the form 0,⋯,0,xn,0⋯ for every n∈ℤ+ is hypercyclic(iv)
limsupn⟶∞∏k=1nωn=+∞Note that in [7], item (iii) is written: “The restriction T:Y⟶Y to any T-invariant (separable) closed subspace Y⊂ℓ2H is hypercyclic.” But it is not correct because the subspace Y0 consisting of vectors x0,0,0,⋯,x0∈H is invariant but the restriction of T to any separable subspace of Y0 is not hypercyclic since Y0⊂kerT.In Section2, we consider abstract shift similar operators on nonseparable function Hilbert spaces ℓ2H. In particular, abstract differentiation operators and dual operators to abstract multiplication operators can be considered as abstract shift similar operators. In Section 3, we construct examples of topologically transitive operators.


## 2. Abstract Shift Similar Operators

LetHnn=0∞ be a sequence of Hilbert spaces. Throughout this paper, we assume that all Hn are nontrivial; that is, Hn≠0 and not necessary separable. Let us suppose that for every n and m,Hn is isomorphic to Hm. We denote by ℓ2Hn=ℓ2Hnn=0∞ the Hilbert space consisting of elements x=x0,x1,⋯,xn,⋯,xk∈Hk endowed with norm x=∑i=0∞xi21/2. Let ωn be a sequence of positive numbers (weights). Let us fix a sequence of isomorphisms Jm:Hm⟶Hm−1,Jm=1,m∈ℕ. An operator T:ℓ2Hn⟶ℓ2Hn will be called a backward weighted shift (with respect to the familyJm) with weight sequenceωn if it is of the following form:
(5)Tx=ω1J1x1,ω2J2x2,⋯,ωmJmxm,⋯.From Proposition4, it follows the next corollary.Corollary 5.
LetHnn=0∞ be a sequence of Hilbert spaces and T:ℓ2Hn⟶ℓ2Hn be a backward weighted shift with respect to Jm and with positive weight sequence ωn. Let us suppose that
(6)supm∈ℤ+∏n=0mJn−1<∞.
Then, the following are equivalent:(i)
T is topologically transitive(ii)
There exists a nontrivialT-invariant (separable) closed subspace Y⊂ℓ2Hn on which the restriction of T to Y,T:Y⟶Y, is hypercyclic(iii)
The restrictionT:Y⟶Y to any T-invariant (separable) closed subspace Y⊂ℓ2Hn which contains nonzero vectors of the form 0,⋯,0,xn,0,⋯,xn∈Hn for every n∈ℤ+, is hypercyclic(iv)
limsupn⟶∞∏k=1nωk=∞Proof.
We setH=H0. Then,
(7)J1H1=H0=H,J1∘J2H2=H0=H,J1∘⋯∘JnHn=H0=H,and so on. For any x=x1,⋯,xn,⋯∈ℓ2Hn,(8)J:x↦x0,J1x1,⋯,J1∘⋯∘Jnxn,⋯∈ℓ2H,Jx≤x.
Because of Equation (6), the inverse operator
(9)J−1:z↦z0,J1−1z1,⋯,Jn−1∘⋯∘J1−1zn,⋯∈ℓ2Hn,z∈ℓ2His well defined and bounded. So, J is an isometric isomorphism from ℓ2Hn to ℓ2H. For a closed subspace Y⊂ℓ2Hn, the range JY is a closed subspace of ℓ2H. Let T′=J∘T∘J−1. Then, T′ is a backward weighted shift on ℓ2H with positive weight sequence ωn. Since J is an isomorphism, the topological transitivity of T or hypercyclicity of T on Y is equivalent to the topological transitivity of T′ on ℓ2H, or hypercyclicity of T′ on JY, respectively. Hence, we can apply Theorem 3 for T′.
Note that the case whenJn=νn>0 and supnνn<∞ can be reduced to our case if we consider Jn′=Jn/νn and ωn′=ωnνn instead of Jn and ωn,n∈ℕ, respectively.
An operatorB:ℓ2Hn⟶ℓ2Hn is a backward weighted shift similar operator if there exists an isomorphism A:ℓ2Hn⟶ℓ2En for some sequence of Hilbert spaces En and a backward weighted shift T:ℓ2En⟶ℓ2En such that B=ATA−1. It is clear that B is topologically transitive if and only if T is topologically transitive.
Let us suppose thatHn=Hn′⊕Hn0,n∈ℤ+ and the isomorphisms Jn:Hn⟶Hn−1, be such that Jn is an isometric isomorphism of Hn′ onto Hn−1′ and Jn maps Hn0 onto Hn−10. Also, we denote by πn the projection of Hn onto Hn′. Then, we define a backward weighted partial shift (with respect to family Jm∘πm) with weight sequence ωn on ℓ2Hn by the following:
(10)Tx=ω1J1∘π1x1,ω2J2∘π2x2,⋯,ωmJm∘πmxm,⋯.
In other words,ℓ2Hn=ℓ2Hn′⊕ℓ2Hn0, and T is a backward weighted shift on ℓ2Hn′ and kerT=ℓ2Hn0.
Let us recall that thedensitydX of a metric space X is the smallest cardinality of a dense subset of X. It is well known that two infinite dimensional Hilbert spaces H and E are isometrically isomorphic if and only if dH=dE. The next proposition shows that the backward weighted partial shift can be reduced to the weighted shift.Proposition 6.
LetT be a backward weighted partial shift with respect to the family of operators Jm∘πmm=1∞ with weight sequence ωn on ℓ2Hn. Let us suppose that all Hn are infinite dimensional for n>1 and dHn0≤dHn. Then, T is a backward weighted shift similar operator; that is, there is a sequence of Hilbert spaces Enn=0∞ and an isometric isomorphism I:ℓ2Hn⟶ℓ2En such that I∘T∘I−1 is a backward weighted shift.Proof.
Let us define spacesEn by the following way:
(11)E0=H0⊕H10⊕H20⊕⋯⊕Hm0⊕⋯,En=Hn′,n>1.
Since spacesHn are isomorphic each to others, dHn=dHm for all n,m∈ℤ+. Since dHn0≤dHn, we have dEn=dEm and so En and Em are isomorphic for all n,m∈ℤ+. It is clear that ℓ2Hn and ℓ2En are just different representations of the same space, and so I is the identical operator. Also, the restrictions of Jm∘πm to Em,m>1 are isometric isomorphisms. So, I∘T∘I−1 is a backward weighted shift with weight sequence ωn on ℓ2En.Corollary 7.
LetT:ℓ2Hn⟶ℓ2Hn be a backward weighted partial shift with positive weight sequence ωn which satisfies conditions of Proposition 6. Then, the following are equivalent:
(i)
T is topologically transitive(ii)
There exists a nontrivialT-invariant (separable) closed subspace Y⊂ℓ2Hnn=0∞ on which the restriction of T to Y,T:Y⟶Y, is hypercyclic(iii)
The restrictionT:Y⟶Y to any T-invariant (separable) closed subspace Y⊂ℓ2Hnn=0∞ which contains nonzero vectors of the form 0,⋯,0,xn,0⋯,xn∈Hn′ for every n∈ℤ+, is hypercyclic(iv)
limsupn⟶∞∏k=1nωk=∞.This approach can be generalised by the following way.Theorem 8.
LetD be a surjective bounded operator on a ℓ2-sum ℓ2Hn of Hilbert spaces Hn such that D:Hn⟶Hn−1,n≥0. Then, D is a backward weighted shift similar operator with some weight sequence ωn,0<ωn≤Dn, where Dn is the restriction of D to Hn.Proof.
Let us define the following spaces:(12)H00=H0,H10=kerD1,H11=H1⊖kerD1,H20=kerD2,H21=kerD1∘D2⊖H20,H22=H2⊖kerD1∘D2,Hn0=kerDn,⋯,Hnk=kerDk∘⋯∘Dn−1∘Dn⊖Hnk−1,⋯,Hnn=Hn⊖kerD1∘⋯∘Dn−1∘Dn.
Let us denote by the following:(13)Hk=ℓ2Hnkn≥k=Hkk⊕Hk+1k⊕⋯,k∈ℤ+.
In other words, if0≠x∈Hk, then 0≠D1∘⋯∘Dkx∈H0. Let Dnk be the restriction of Dn to Hnk. Then, Dnk≤Dn≤D,k≤n. Thus, the action of D on Hk can be written by the following:
(14)Dk≔D0k⊕D1k⊕⋯⊕Dnk⊕⋯,k∈ℤ+and Dk=supnDnk. So, Dnk≤Dk≤D,k≤n. It is clear that the spaces ℓ2Hn and ℓ2Hn consist of the same vectors and the identity map I:ℓ2Hn⟶ℓ2Hn is an isometric isomorphism. Thus, Dk is the restriction of IDI−1 to Hk. By the construction, Dk:Hk⟶Hk−1,k>0 and D0=0. Moreover, since kerIDI−1=H0,kerDk=0 if k>0. In addition, since D is onto, Dk must be onto for every k, because the preimage of Hk−1 under Dk,k>0 is Hk. Hence, every Dk,k>0 is an isomorphism.
Let us set the following:(15)Jk≔DkDk,ωk=Dk,fork>1.
Then for everyx=x0,x1,⋯,xn,⋯∈ℓ2Hn,(16)IDI−1x=ω1J1x1,ω2J2x2,⋯,ωnJnxn,⋯,that is, IDI−1 is a backward weighted shift.
Let us show thatωk>0 for every k∈ℕ. Since all spaces Hk are nontrivial and D is onto, there is a vector a1∈H1 such that a1∉kerD. So, a1∈H1 and D1a1=D1a1≠0; that is, ω1=D1≠0. If already we have nonzero elements aj∈Hj∩Hj,j=1,2,⋯,k−1, then we denote by ak a vector in Hk in the preimage of ak−1 under Dk. By the assumption ak−1∉kerD1∘⋯∘Dk−1, so ak∉kerD1∘⋯∘Dk. Thus, ak∈Hk and Dkak=Dkak=ak−1≠0. Hence, ωk=Dk≠0.
An elementx∈ℓ2Hn is a finite type vector if there is m∈ℕ such that xk=0 for all k≥m.Definition 9.
LetHn,n∈ℤ+ be the Hilbert spaces. We say that ℓ2Hn is a graded Hilbert space, if there is a bilinear map x,y↦xy (multiplication) defined for every finite type vector x and an arbitrary y∈ℓ2Hn with values in ℓ2Hn such that the multiplication is associative, nondegenerated (that is, xy=0 only if x=0 or y=0) and xnym∈Hn+m for every xn∈Hn and ym∈Hm.Definition 10.
For a given graded Hilbert spaceℓ2Hn and a finite type vector a∈ℓ2Hn, we denote by Ma the multiplication operator.
(17)Max=ax,x∈ℓ2Hn.Theorem 11.
Leta∈Hm and Ma be continuous on ℓ2Hn. If the dual operator Ma∗:ℓ2Hn∗=ℓ2Hn∗⟶ℓ2Hn∗ is surjective, then it is a backward weighted shift similar operator.Proof.
Let(18)E0=H0⊕⋯⊕Hm−1,⋯,Ek=Hkm⊕⋯⊕Hkm+k−1,⋯and Hn=En∗ be the Hermitian dual space for every n∈ℤ+. Then, Ma maps En to En+1,n∈ℤ+ and Ma∗ maps Hn to Hn−1,n>0, and H0=kerMa∗. Let Ma∗n be the restriction of Ma∗ to Hn. Since Ma is continuous, Ma∗ is continuous on ℓ2Hn as well and supnMa∗n<∞. We set the following:
(19)Jn=Ma∗nMa∗n,ωn=Ma∗n.
SinceMa∗ is surjective, Ma∗n are surjective for all n∈ℤ+ and since H0=kerMa∗,Ma∗n are bijective for n>0. So,
(20)Ma∗x0,x1,⋯,xk,⋯=ω1J1x1,ω2J2x2,⋯,ωkJkxk,⋯is a backward weighted shift on ℓ2Hn.
Note that ifa∈H1, then Ek=Hk and ωn=Ma∗n=Man−1,, where Mak is the restriction of Ma to Hk,k∈ℤ+.Definition 12.
Letℓ2Hn be a graded Hilbert space. A continuous linear operator D:ℓ2Hn⟶ℓ2Hn is called a differential operator if H0⊂kerD, the restriction of D to Hn maps Hn onto Hn−1,n>1 and
(21)Dxy=Dxy+xDyfor every finite type vector x and for all y∈ℓ2Hn.Corollary 13.
LetD be a differential operator on a graded Hilbert space ℓ2Hn such that D is onto and D1≠0. Then, there is a sequence of Hilbert spaces Hnn=0∞ and an isometric isomorphism I:ℓ2Hn⟶ℓ2Hn such that I∘D∘I−1 is a backward weighted shift with some weight sequence ωn,0<ωn≤Dn.Proof.
Since∥D1∥≠0, there is a∈H1 such that Da≠0. So, a∈H1. Moreover, for every k∈ℕ,ak=a⋯a⏟k≠0 and Dak=kak−1. Thus, ak∈Hk and we can apply Theorem 2.4.Corollary 14.
LetD be a continuous surjective differential operator on a graded Hilbert space ℓ2Hn. The following are equivalent:
(i)
D is topologically transitive(ii)
There exista∈H1,ε>0, and m∈ℕ such that Da≠0 and(22)limsupm⟶∞∏n=1mnan−1an=∞.(iii)(23)limsupm⟶∞∏n=1mDn=∞.Proof.
As we observed,ak∈Hk and so
(24)ωk=Dk≥Dkakak=kak−1ak.
By Theorem8., ωk>0,k∈ℕ. So if ωk>1+ε for all k>m, then
(25)∏n=1∞ωn=∞and D must be topologically transitive.


## 3. Examples of Topologically Transitive Operators

### 3.1. Partial Shifts

It is well known that any elementx of the (complete) Hilbert tensor product ℓ2⊗hℓ2 can be represented by the following series
(26)x=∑i,j=0∞xijei⊗ej,whereeii∈ℤ+isthestandardbasisinℓ2or by the following infinite matrix
(27)x=xiji,j∈ℤ+=x00x01⋯x0n⋯x10x11⋯x1n⋯⋯⋯⋯⋯⋯xn0xn1⋯xnn⋯⋯⋯⋯⋯⋯with x2=∑i,j=0∞xij2. Note that ℓ2⊗hℓ2 can be considered as ℓ2ℓ2 by the following representation:
(28)x=x0,x1,⋯,xn,⋯∈ℓ2ℓ2,wherexn=xn0,xn1,⋯,xnn,⋯∈ℓ2,n∈ℤ+.Letω=ωiji,j be an infinite matrix of positive numbers. We consider the following operators on ℓ2ℓ2. Let
(29)Tωx=ωi+1,jxi+1,ji,j∈ℤ+=ω10x10ω11x11⋯ω1nx1n⋯ω20x20ω21x21⋯ω2nx2n⋯⋯⋯⋯⋯⋯ωn0xn0ωn1xn1⋯ωnnxnn⋯⋯⋯⋯⋯⋯and Tωt be the transposed operator, Tωx=ωi,j+1xi,j+1i,j∈ℤ+. Let ωi=supjωij. From Theorem 8., it follows that if Tω is continuous and
(30)limsupn⟶∞∏i=1nωi=+∞,then operatorsTωmTωtn are topologically transitive for every m,n,m+1≥0. In particular, the operator
(31)TωTωtx=ω11x11ω12x12⋯ω1nx1n⋯ω21x21ω22x22⋯ω2nx2n⋯⋯⋯⋯⋯⋯ωn1xn1ωn2xn2⋯ωnnxnn⋯⋯⋯⋯⋯⋯looks like weighted diagonal shift operator. It is clear that these examples can be generalised for any tensor degree ⊗hmℓ2,m∈ℕ. Next, we consider a nonseparable case.LetA be the set of indexes (uncountable, in general) which can be represented as a disjoint union of subsets Ai.
(32)A=∐i=0∞Ai,and there are a sequence of bijections τn:An⟶An−1,n∈ℕ. So, we can write ℓ2A=ℓ2ℓ2An and define Jn:ℓ2An⟶ℓ2An−1,n∈ℕ by Jnxi=xτni,i∈An. By Corollary 5, we have that
(33)Tx=ω1J1x1,ω2J2x2,⋯,ωmJmxm,⋯is topologically transitive if and only if the weight ωm satisfies Equation (31). Moreover, the Hilbert tensor product ℓ2A⊗hℓ2A is isomorphic to
(34)ℓ2A×A=ℓ2ℓ2An×Am.For a given weight matrix ω=ωi,j,i,j∈ℤ+,ωi,j>0, we define operators Tω and Tωt on ℓ2ℓ2An×Am by the following:
(35)Tωxnm=xτnmandTωtxnm=xnτm,wherexnm∈ℓ2An×Am.Corollary 15.
Let(36)supi,j∈ℤ+ωi,j<∞,
then for everyk,j∈ℤ+,k+j>0, operators TωkTωtj are topologically transitive if and only if
(37)limsupn⟶∞∏i=1nsupmωmi=+∞.


### 3.2. Differential Operator on a Function Space

Examples of Hilbert spaces of analytic functions with continuous differential operators can be found among reproducing kernel Hilbert spaces.Definition 16.
LetZ be an abstract set and H a Hilbert space of complex valued functions f on Z equipped with inner product ·∣·H. A function Kx∣z defined on Z×Z is called reproducing kernel of a closed subspace HK⊂H if
(i)
For any fixedz∈Z,Kx∣z belongs to HK as a function in x∈Z(ii)
For anyf∈HK and for any z∈Z,fz=f∣K·∣zH
The spaceHK is called a Hilbert space with reproducing kernel. A function h:Z⟶H such that
(38)fx=f·∣hxHfor every f∈HK and x∈Z is called the kernel function of HK.
LetHγ be the Hilbert space of power series
(39)fz=∑n=0∞f^nzn,z∈ℂendowed with the norm
(40)f2=∑n=0∞γn−2f^n2≤∞,where γ=γn is a positive sequence and f^n∈ℂ. So, the inner product on Hγ is defined by f∣g=∑n=0∞γn−2f^ng^n and Hγ is a reproducing kernel space with the reproducing kernel.
(41)Kx∣z=∑n=0∞γn2xnz¯n.
It is known [16] that if γn+1/γn⟶0 as n⟶∞,Hγ consists of entire functions on ℂ.
LetD=d/dz be the differential operator on Hγ.Then, Dnf^nzn=f^nnzn−1,n>0, and so Dn=nγn/γn−1. Thus, D is continuous (c. f. [9]) if and only if
(42)supnDn=supnnγnγn−1<∞.
On the other hand, according to Corollary14, D is topologically transitive and so hypercyclic if and only if
(43)limsupm⟶∞∏n=1mnγnγn−1⟶∞.
It is clear thatH∗γ=H1/γ. Let us consider the multiplication operator Mz on Hγ,Mzf=zfz. Then, Mzn=γn/γn+1. According to Theorem 11, Mz is bounded if and only if supnγn/γn+1<∞ and Mz is topologically transitive if and only if
(44)limsupm⟶∞∏n=1mγnγn+1⟶∞.
This result agrees with [8]. Various Hilbert spaces of analytic functions of infinite many variables were constructed in [14, 17, 18] using the Hilbertian symmetric tensor products of Hilbert spaces. Such constructions can be easily extended to nonseparable Hilbert spaces. In [14], the authors found some conditions under which the translation operator fx↦fx+a on a Hilbert space of entire analytic functions on ℓ2 is hypercyclic. Taking into account that the translation operator can be represented as an exponential function of a differential operator, we would like to ask the following question.Question 17.
LetD be a differential operator on a graded nonseparable Hilbert space. Suppose that D is topologically transitive. Does
(45)eD=∑n=0∞Dnn!topologically transitive?


## 3.1. Partial Shifts

It is well known that any elementx of the (complete) Hilbert tensor product ℓ2⊗hℓ2 can be represented by the following series
(26)x=∑i,j=0∞xijei⊗ej,whereeii∈ℤ+isthestandardbasisinℓ2or by the following infinite matrix
(27)x=xiji,j∈ℤ+=x00x01⋯x0n⋯x10x11⋯x1n⋯⋯⋯⋯⋯⋯xn0xn1⋯xnn⋯⋯⋯⋯⋯⋯with x2=∑i,j=0∞xij2. Note that ℓ2⊗hℓ2 can be considered as ℓ2ℓ2 by the following representation:
(28)x=x0,x1,⋯,xn,⋯∈ℓ2ℓ2,wherexn=xn0,xn1,⋯,xnn,⋯∈ℓ2,n∈ℤ+.Letω=ωiji,j be an infinite matrix of positive numbers. We consider the following operators on ℓ2ℓ2. Let
(29)Tωx=ωi+1,jxi+1,ji,j∈ℤ+=ω10x10ω11x11⋯ω1nx1n⋯ω20x20ω21x21⋯ω2nx2n⋯⋯⋯⋯⋯⋯ωn0xn0ωn1xn1⋯ωnnxnn⋯⋯⋯⋯⋯⋯and Tωt be the transposed operator, Tωx=ωi,j+1xi,j+1i,j∈ℤ+. Let ωi=supjωij. From Theorem 8., it follows that if Tω is continuous and
(30)limsupn⟶∞∏i=1nωi=+∞,then operatorsTωmTωtn are topologically transitive for every m,n,m+1≥0. In particular, the operator
(31)TωTωtx=ω11x11ω12x12⋯ω1nx1n⋯ω21x21ω22x22⋯ω2nx2n⋯⋯⋯⋯⋯⋯ωn1xn1ωn2xn2⋯ωnnxnn⋯⋯⋯⋯⋯⋯looks like weighted diagonal shift operator. It is clear that these examples can be generalised for any tensor degree ⊗hmℓ2,m∈ℕ. Next, we consider a nonseparable case.LetA be the set of indexes (uncountable, in general) which can be represented as a disjoint union of subsets Ai.
(32)A=∐i=0∞Ai,and there are a sequence of bijections τn:An⟶An−1,n∈ℕ. So, we can write ℓ2A=ℓ2ℓ2An and define Jn:ℓ2An⟶ℓ2An−1,n∈ℕ by Jnxi=xτni,i∈An. By Corollary 5, we have that
(33)Tx=ω1J1x1,ω2J2x2,⋯,ωmJmxm,⋯is topologically transitive if and only if the weight ωm satisfies Equation (31). Moreover, the Hilbert tensor product ℓ2A⊗hℓ2A is isomorphic to
(34)ℓ2A×A=ℓ2ℓ2An×Am.For a given weight matrix ω=ωi,j,i,j∈ℤ+,ωi,j>0, we define operators Tω and Tωt on ℓ2ℓ2An×Am by the following:
(35)Tωxnm=xτnmandTωtxnm=xnτm,wherexnm∈ℓ2An×Am.Corollary 15.
Let(36)supi,j∈ℤ+ωi,j<∞,
then for everyk,j∈ℤ+,k+j>0, operators TωkTωtj are topologically transitive if and only if
(37)limsupn⟶∞∏i=1nsupmωmi=+∞.


## 3.2. Differential Operator on a Function Space

Examples of Hilbert spaces of analytic functions with continuous differential operators can be found among reproducing kernel Hilbert spaces.Definition 16.
LetZ be an abstract set and H a Hilbert space of complex valued functions f on Z equipped with inner product ·∣·H. A function Kx∣z defined on Z×Z is called reproducing kernel of a closed subspace HK⊂H if
(i)
For any fixedz∈Z,Kx∣z belongs to HK as a function in x∈Z(ii)
For anyf∈HK and for any z∈Z,fz=f∣K·∣zH
The spaceHK is called a Hilbert space with reproducing kernel. A function h:Z⟶H such that
(38)fx=f·∣hxHfor every f∈HK and x∈Z is called the kernel function of HK.
LetHγ be the Hilbert space of power series
(39)fz=∑n=0∞f^nzn,z∈ℂendowed with the norm
(40)f2=∑n=0∞γn−2f^n2≤∞,where γ=γn is a positive sequence and f^n∈ℂ. So, the inner product on Hγ is defined by f∣g=∑n=0∞γn−2f^ng^n and Hγ is a reproducing kernel space with the reproducing kernel.
(41)Kx∣z=∑n=0∞γn2xnz¯n.
It is known [16] that if γn+1/γn⟶0 as n⟶∞,Hγ consists of entire functions on ℂ.
LetD=d/dz be the differential operator on Hγ.Then, Dnf^nzn=f^nnzn−1,n>0, and so Dn=nγn/γn−1. Thus, D is continuous (c. f. [9]) if and only if
(42)supnDn=supnnγnγn−1<∞.
On the other hand, according to Corollary14, D is topologically transitive and so hypercyclic if and only if
(43)limsupm⟶∞∏n=1mnγnγn−1⟶∞.
It is clear thatH∗γ=H1/γ. Let us consider the multiplication operator Mz on Hγ,Mzf=zfz. Then, Mzn=γn/γn+1. According to Theorem 11, Mz is bounded if and only if supnγn/γn+1<∞ and Mz is topologically transitive if and only if
(44)limsupm⟶∞∏n=1mγnγn+1⟶∞.
This result agrees with [8]. Various Hilbert spaces of analytic functions of infinite many variables were constructed in [14, 17, 18] using the Hilbertian symmetric tensor products of Hilbert spaces. Such constructions can be easily extended to nonseparable Hilbert spaces. In [14], the authors found some conditions under which the translation operator fx↦fx+a on a Hilbert space of entire analytic functions on ℓ2 is hypercyclic. Taking into account that the translation operator can be represented as an exponential function of a differential operator, we would like to ask the following question.Question 17.
LetD be a differential operator on a graded nonseparable Hilbert space. Suppose that D is topologically transitive. Does
(45)eD=∑n=0∞Dnn!topologically transitive?


## 4. Conclusion

We can see that dual to abstract multiplication operatorsMa∗ and abstract differentiation operators D on Hilbert spaces can be represented as backward weighted shift operators. Thus, we can use known results about topological transitivity of backward weighted shift operators to get conditions of topological transitivity of operators Ma∗ and D.



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