arXiv subjects
Publications and source records attributed to B. P. Duggal.
The paper extends previous results on the nth root problem to a large class of Hilbert-space operators, namely, the class of all normaloid operators with normaloid parts, which includes the paranormal operators, and also the $k$-paranormal operators. It is shown that if a normaloid operator with normaloid parts has a normal nth power, then it is normal.
The paper proves two results involving a pair (A,B) of P-biisometric or (m,P)-biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair (A,B) is (m,P)-biisometric for some m, then it is a P-biisometric pair. The important case when P is invertible is treated in detail. It is also shown that if (A,B) is P-biisometric, then there are biorthogonal sequences with respect to the inner product <.;.>_P= that have a shift-like behaviour with respect to this inner product.
It is known that supercyclicity implies strong stability. It is not known whether weak l-sequential supercyclicity implies weak stability. In this paper we prove that weak l-sequential supercyclicity implies weak quasistability. Corollaries concerning the characterisation of (i) weakly l-sequentially supercyclic vectors that are not (strongly) supercyclic, and (ii) weakly l-sequentially supercyclic isometries, are also proved.
Given commuting $d$-tuples $\mathbb{S}_i$ and $\mathbb{T}_i$, $1\leq i\leq 2$, Banach space operators such that the tensor products pair $(\mathbb{S}_1\otimes\mathbb{S}_2,\mathbb{T}_1\otimes\mathbb{T}_2)$ is strict $m$-isometric (resp., $\mathbb{S}_1$, $\mathbb{S}_2$ are invertible and $(\mathbb{S}_1 \otimes \mathbb{S}_2, \mathbb{T}_1 \otimes\mathbb{T}_2)$ is strict $m$-symmetric), there exist integers $m_i >0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $(\mathbb{S}_1, {\frac{1}{c}}\mathbb{T}_1)$ is strict $m_1$-isometric and $(\mathbb{S}_2, c\mathbb{T}_2)$ is strict $m_2$-isometric (resp., there exist integers $m_i >0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $(\mathbb{S}_1,{\frac{1}{c}}\mathbb{T}_1)$ is strict $m_1$-symmetric and $(\mathbb{S}_2, c\mathbb{T}_2)$ is strict $m_2$-symmetric. However, $(\mathbb{S}_i,\mathbb{T}_i)$ is strict $m_i$-isometric (resp., strict $m_i$-symmetric) for $1\leq i\leq 2$ implies only that $(\mathbb{S}_1\otimes \mathbb{S}_2, \mathbb{T}_1\otimes \mathbb{T}_2)$ is $m$-isometric (resp., $(\mathbb{S}_1 \otimes \mathbb{S}_2, \mathbb{T}_1\otimes\mathbb{T}_2)$ is $m$-symmetric).
There is no supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}.$ There exist, however, weakly l-sequentially supercyclic unitary operators$.$ We show that if $T$ is a weakly l-sequentially supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}$, then it has an extension $\widehat T$ which is a weakly l-sequentially supercyclic singular-continuous unitary (and $\widehat T$ has a Rajchman scalar spectral measure whenever $T$ is weakly stable)$.$ The above result implies $σ_{\kern-1ptP}(T)=σ_{\kern-1ptP}(T^*)=\varnothing$, and also that if a weakly l-sequentially supercyclic operator is similar to an isometry, then it is similar to a unitary operator.
This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.
This paper provides conditions (i) to distinguish weak supercyclicity form supercyclicity for operators acting on normed and Banach spaces, and also (ii) to ensure when weak supercyclicity implies weak stability.
A Hilbert space operator $T\in B$ is $(m,P)$-expansive, for some positive integer $m$ and operator $P\in B$, if $\sum_{j=0}^m{(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)T^{*j}PT^j}\leq 0$. No Drazin invertible operator $T$ can be $(m,I)$-expansive, and if $T$ is $(m,P)$-expansive for some positive operator $P$, then necessarily $P$ has a decomposition $P=P_{11}\oplus 0$. If $T$ is $(m,|T^n|^2)$-expansive for some positive integer $n$, then $T^n$ has a decomposition $T^n=\left(\begin{array}{clcr}U_1P_1 & X\\0 & 0\end{array}\right)$; if also $\left(\begin{array}{clcr}I_1 & X\\X^* & X^*X\end{array}\right)\geq I$, then $\left(\begin{array}{clcr}P_1U_1 & P_1X\\0 & 0\end{array}\right)$ is $(m,I)$-expansive and $\left(\begin{array}{clcr}P^{\frac{1}{2}}_1U_1P^{\frac{1}{2}}_1 & P_1^{\frac{1}{2}}X\\0 & 0\end{array}\right)$ is $(m,I)$-expansive in an equivalent norm on $H$.
Given Hilbert space operators $P,T\in B(\H), P\geq 0$ invertible, $T$ is $(m,P)-$ expansive (resp., $(m,P)-$ isometric) for some positive integer $m$ if $\triangle_{T^*,T}^m(P)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right){T^*}^jPT^j\leq 0$ (resp., $\triangle_{T^*,T}^m(P)=0$). An $(m,P)-$ expansive operator $T$ is power bounded if and only if it is a $C_{1\cdot}-$ operator which is similar to an isometry and satisfies $\triangle_{T^*,T}^n(Q)=0$ for some positive invertible operator $Q\in B(\H)$ and all integers $n\geq 1$. If, instead, $T$ is an algebraic $(m,I)-$ expansive operator, then either the spectral radius $r(T)$ of $T$ is greater than one or $T$ is the perturbation of a unitary by a nilpotent such that $T$ is $(2n-1, I)-$ isometric for some positive integers $m_0 \leq m$, $m_0$ odd, and $n \geq \frac{m_0 +1}{2}$.
Given Hilbert space operators $A_i, B_i$, $i=1,2$, and $X$ such that $A_1$ commutes with $A_2$ and $B_!$ commutes with $B_2$, and integers $m, n\geq 1$, we say that the pairs of operators $(B_1,A_1)$ and $(B_2,A_2)$ are left-$(X, (m,n))$-symmetric, denoted $((B_1,A_1),(B_2,A_2))\in {\rm left}-(X,(m,n))-{\rm symmetric}$ if $$ \sum_{j=0}^m\sum_{k=0}^n (-1)^{j+k}\left(\begin{array}{clcr}m\\j\end{array}\right) \left(\begin{array}{clcr}n\\k\end{array}\right) B_1^{m-j}B_2^{n-k} X A_2^{n-k}A_1^{j}=0.$$An important class of left-$(X,(m,n))-$symmetric operators is obtained uponchoosing $B_1=B_2=A^*_1=A^*_2=A^*$ and $X=I$: such operators have been called $(m,n)-$isosymmetric, and a study of the spectral picture and maximal invariant subspaces of $(m,n)-$isosymmetric operators has been carried out by Stankus \cite{St}. The current work considers stability under perturbations by commuting nilpotents, and products of commuting, left-$(X, (m,n))-$symmetric operators. It is seen that $(X, (m,n))-$isosymmetric Drazin invertible operators $A$ have a particularly interesting structure.
Every new inner product in a Hilbert space is obtained from the original one by means of a unique positive operator$.$ The first part of the paper is a survey on applications of such a technique, including a characterization of similarity to isometries$.$ The second part focuses on Banach limits for dealing with power bounded operators. It is shown that if a power bounded operator for which the sequence of shifted Cesàro means converges (at least in the weak topology) uniformly in the shift parameter, then it has a Cesàro asymptotic limit coinciding with its $φ$-asymptotic limit for all Banach limits $φ$.
Given Hilbert space operators $T, S\in\B$, let $\triangle$ and $δ\in B(\B)$ denote the elementary operators $\triangle_{T,S}(X)=(L_TR_S-I)(X)=TXS-X$ and $δ_{T,S}(X)=(L_T-R_S)(X)=TX-XS$. Let $d=\triangle$ or $δ$. Assuming $T$ commutes with $S^*$, and choosing $X$ to be the positive operator $S^{*n}S^n$ for some positive integer $n$, this paper exploits properties of elementary operators to study the structure of $n$-quasi $[m,d]$-operators $d^m_{T,S}(X)=0$ to bring together, and improve upon, extant results for a number of classes of operators, amongst them $n$-quasi left $m$-invertible operators, $n$-quasi $m$-isometric operators, $n$-quasi $m$-selfadjoint operators and $n$-quasi $(m,C)$ symmetric operators (for some conjugation $C$ of $\H$). It is proved that $S^n$ is the perturbation by a nilpotent of the direct sum of an operator $S_1^n=(S|_{\overline{S^n(\H)}})^n$ satisfying $d^m_{T_1,S_1}(I_1)=0$, $T_1=T|_{\overline{S^n(\H)}}$, with the $0$ operator; if also $S$ is left invertible, then $S^n$ is similar to an operator $B$ such that $d^m_{B^*,B}(I)=0$. For power bounded $S$ and $T$ such that $ST^*-T^*S=0$ and $\triangle_{T,S}(S^{*n}S^n)=0$, $S$ is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators $T, S$ satisfying $d^m_{T,S}(I)=0$, given certain commutativity properties, transfers to operators satisfying $S^{*n}d^m_{T,S}(I)S^n=0$.
We use elementary algebraic properties of left, right multiplication operators to prove some deep structural properties of left $m$-invertible, $m$-isometric, $m$-selfadjoint and other related classes of Banach space operators, often adding value to extant results.
A Hilbert space operator $A\in\B$ is left $(X,m)$-invertible by $B\in\B$ (resp., $B\in\B$ is an $(X,m)$-adjoint of $A\in\B$) for some operator $X\in\B$ if $\triangle_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{m-j}XA^{m-j}=0$ (resp., $δ_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{(m-j)}XA^j=0$). No Drazin invertible operator $A\in\B$, with Drazin inverse $A_d$, can be left $(I,m)$-invertible (equivalently, $m$-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin inverrtible operators $A$, it is seen that the existence of an $X$ acts as a conduit for implications $\triangle_{B,A}(X)=0\Longrightarrow δ^m_{C,A}(X)=0$, where the pair $(B,C)=$ either $(A,A_d)$ or $(A_d,A)$ or $(A^*,A^*_d)$ or $(A^*_d,A^*)$. Reverse implications fail. Assuming certain commutativity conditions, it is seen that $\triangle_{A^*_d,A}^m(X)=0=\triangle^n_{B^*_d,B}(Y)$ implies $δ^{m+n-1}_{A^*B^*,AB}(XY)=0=δ^{m+n-1}_{A^*+B^*,A+B}(XY)$.
A Drazin invertible Hilbert space operator $T\in \B$, with Drazin inverse $T_d$, is $(n,m)$-power D-normal, $T\in [(n,m) DN]$, if $[T_d^n,T^{*m}]=T^n_dT^{*m}-T^{*m}T_d^n=0$; $T$ is $(n,m)$-power D-quasinormal, $T\in [(n,m) DQN]$, if $[T_d^n,T^{*m}T]=0$. Operators $T\in [(n,m) DN]$ have a representation $T=T_1\oplus T_0$, where $T_1$ is similar to an invertible normal operator and $T_0$ is nilpotent. Using this representation, we have a keener look at the structure of $[(n,m) DN]$ and $[(n,m) DQN]$ operators. It is seen that $T\in [(n,m) DN]$ if and only if $T\in [(n,m) DQN]$, and if $[T,X]=0$ for some operators $X\in\B$ and $T\in [(1,1) DN]$, then $[T^*_d,X]=0$. Given simply polar operators $S, T\in [(1,1) DN]$ and an operator $A=\left(\begin{array}{clcr} T&C 0&S \end{array}\right) \in B(\H\oplus\H)$, $A\in [(1,1) DN]$ if and only if $C$ has a representation $C=0\oplus C_{22}$.
For $n$-normal operators $A$ [2, 4, 5], equivalently $n$-th roots $A$ of normal Hilbert space operators, both $A$ and $A^*$ satisfy the Bishop--Eschmeier--Putinar property $(β)_ε$, $A$ is decomposable and the quasi-nilpotent part $H_0(A-λ)$ of $A$ satisfies $H_0(A-λ)^{-1}(0)=(A-λ)^{-1}(0)$ for every non-zero complex $λ$. $A$ satisfies every Weyl and Browder type theorem, and a sufficient condition for $A$ to be normal is that either $A$ is dominant or $A$ is a class ${\mathcal A}(1,1)$ operator.
A Hilbert space operator $S\in\B$ is $n$-quasi left $m$-invertible (resp., left $m$-invertible) by $T\in\B$, $m,n \geq 1$ some integers, if $S^{*n}p(S,T)S^n=0$ (resp., $p(S,T)=0$), where $p(S,T)=\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right)T^jS^j}$. Left $m$-invertible and $n$-quasi left $m$-invertible operators share a number of properties. Thus, if $S$ is $n$-quasi left $m$-invertible, then $S^n$ is the perturbation by a nilpotent of the direct sum of a left $m$-invertible with the $0$ operator. In particular, if $T=S^*$ (so that $S$ is $n$-quasi $m$-isomertric) and $|(S|_{\overline{S^n(\H)}})^n|$ is not the identity operator, then $S^n$ is similar to an $m$-isometry. For a power bounded $n$-quasi left $m$-invertible operator $S$ such that $T$ is (also) power bounded. and $ST^*-T^*S=0$, $S$ is polaroid (i.e., isolated points of the spectrum are poles); the product of an $n$-quasi left $m_1$-invertible operator with a left $m_2$-invertible operator, given certain commutativity properties, is $n$-quasi left $(m_1+m_2-1)$-invertible; again, if $ST^*-T^*S=0$ and $N$ is an $n_1$-nilpotent which commutes with $S$, then $T$ is an $(n+n_1-1)$-quasi left $(m+n_1-1)$-inverse of $S+N_1$. These results have applications to $n$-quasi $m$-isometries \cite{AS}, $[m,C]$-isometries \cite{CKL}, and (left invertible) $m$-symmetric \cite{CLM} and $m$-selfadjoint \cite{L} operators.
A Hilbert space operator $S\in\B$ is left $m$-invertible by $T\in\B$ if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right)T^jS^j}=0,$$ $S$ is $m$-isometric if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right){S^*}^jS^j}=0$$ and $S$ is $(m,C)$-isometric for some conjugation $C$ of $\H$ if $$\sum_{j=0}^m{(-1)^{m-j}\left(\begin{array}{clcr}m\\j\end{array}\right){S^*}^jCS^jC}=0.$$ If a power bounded operator $S$ is left invertible by a power bounded operator $T$, then $S$ (also, $T^*$) is similar to an isometry. Translated to $m$-isometric and $(m,C)$-isometric operators $S$ this implies that $S$ is $1$-isometric, equivalently isometric, and (respectively) $(1,C)$-isometric.