scholarly journals Structure of n-quasi left m-invertible and related classes of operators

2020 ◽  
Vol 53 (1) ◽  
pp. 249-268
Author(s):  
Bhagwati Prashad Duggal ◽  
In Hyun Kim
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points ◽  
Elementary Operators ◽  
Adjoint Operators ◽  
Number Of Classes ◽  
Symmetric Operators ◽  
Power Bounded

AbstractGiven Hilbert space operators T,S\in B( {\mathcal H} ), let \text{Δ} and \delta \in B(B( {\mathcal H} )) denote the elementary operators {\text{Δ}}_{T,S}(X)=({L}_{T}{R}_{S}-I)(X)=TXS-X and {\delta }_{T,S}(X)=({L}_{T}-{R}_{S})(X)=TX-XS. Let d=\text{Δ} or \delta . Assuming T commutes with {S}^{\ast }, and choosing X to be the positive operator {S}^{\ast 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}_{T,S}^{m}(X)=0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m,C) symmetric operators (for some conjugation C of {\mathcal H} ). It is proved that {S}^{n} is the perturbation by a nilpotent of the direct sum of an operator {S}_{1}^{n}={\left(S{|}_{\overline{{S}^{n}( {\mathcal H} )}}\right)}^{n} satisfying {d}_{{T}_{1},{S}_{1}}^{m}({I}_{1})=0, {{T}_{1}=T}_{\overline{{S}^{n}( {\mathcal H} )}}, with the 0 operator; if S is also left invertible, then {S}^{n} is similar to an operator B such that {d}_{{B}^{\ast },B}^{m}(I)=0. For power bounded S and T such that S{T}^{\ast }-{T}^{\ast }S=0 and {\text{Δ}}_{T,S}({S}^{\ast 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}_{T,S}^{m}(I)=0, given certain commutativity properties, transfers to operators satisfying {S}^{\ast n}{d}_{T,S}^{m}(I){S}^{n}=0.

Weyl’s theorem for direct sums

2007 ◽  
Vol 44 (2) ◽  
pp. 275-290
Author(s):  
Bhagwati Duggal ◽  
Carlos Kubrusly
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Weyl’S Theorem ◽  
Direct Sums ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points

Let T and S be Hilbert space operators such that Weyl’s theorem holds for both of them. In general, it does not follow that Weyl’s theorem holds for the direct sum T ⊕ S . We give asymmetric sufficient conditions on T and S to ensure that the direct sum T ⊕ S satisfies Weyl’s theorem. It is assumed that just one of the direct summands satisfies Weyl’s theorem but is not necessarily isoloid, while the other has no isolated points in its spectrum.

Entropy of dynamical systems and perturbations of operators

1991 ◽  
Vol 11 (4) ◽  
pp. 779-786 ◽  
Author(s):  
Dan Voiculescu
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Lower Bound ◽  
Hausdorff Measure ◽  
Spectral Measure ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
Hilbert Space Operators ◽  
Neumann Class ◽  
Adjoint Operators

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

Operator m-convex functions

2018 ◽  
Vol 25 (1) ◽  
pp. 93-107
Author(s):  
Jamal Rooin ◽  
Akram Alikhani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Convex Functions ◽  
Weight Functions ◽  
Operator Inequality ◽  
Jensen Inequality ◽  
Hilbert Space Operators ◽  
Operator Version ◽  
Continuous Fields ◽  
Adjoint Operators ◽  
Comprehensive Study

AbstractThe aim of this paper is to present a comprehensive study of operatorm-convex functions. Let{m\in[0,1]}, and{J=[0,b]}for some{b\in\mathbb{R}}or{J=[0,\infty)}. A continuous function{\varphi\colon J\to\mathbb{R}}is called operatorm-convex if for any{t\in[0,1]}and any self-adjoint operators{A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained inJ, we have{\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuousm-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operatorm-convexity, we extend the Choi–Davis–Jensen inequality for operatorm-convex functions. We also present an operator version of the Jensen–Mercer inequality form-convex functions and generalize this inequality for operatorm-convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operatorm-convex functions, we construct them-Jensen operator functional and obtain an upper bound for it.

scholarly journals Symmetrisable operators :Part III Hilbert space operators Symmetrisable by bounded operators

1964 ◽  
Vol 4 (1) ◽  
pp. 31-48 ◽  
Author(s):  
J. P. O. Silberstein
Keyword(s):  
Hilbert Space ◽  
The Other ◽  
Restricted Class ◽  
Bounded Operators ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Symmetric Operators ◽  
Considerable Complexity

The fact that the most general symmetrisable operators in Hilbert Space do not possess a number of the desirable properties of such operators in unitary spaces makes it necessary to look for a more restricted class of operators. There are two reasons for our particular choice. In the first place many of the conditions introduced in the course of Part II concerned reltionships between the domain of the symmetrising operatorHand the domain and range of the symmetrisable operatorA. These conditions are now all automatically satisfied. The other reason is that the construction used in section 4 to relate symmetrisable operators to certain symmetric operators clearly required that eitherHorH−1was bounded. The case ofH−1bounded has already been dealt with in section 9 and shown to be fairly simple. The case in which H is bounded is clearly of considerable complexity, since we have already seen (example in proof of Theorem 10.6.) that the con |H| the bound ofHby

scholarly journals Structure of Iso-Symmetric Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 256
Author(s):  
Bhagwati Prashad Duggal ◽  
In-Hyoun Kim
Keyword(s):  
Matrix Representation ◽  
Triangular Matrix ◽  
Space Operator ◽  
Left Multiplication ◽  
Direct Sum ◽  
Upper Triangular Matrix ◽  
Symmetric Operators ◽  
Bilateral Shift ◽  
Positive Integers ◽  
Power Bounded

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

A Class of Polynomials in Self-Adjoint Operators in Spaces with an Indefinite Metric

1968 ◽  
Vol 20 ◽  
pp. 673-678 ◽  
Author(s):  
C.-Y. Lo
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Orthogonal Decomposition ◽  
Inner Product ◽  
Indefinite Metric ◽  
Indefinite Inner Product ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Adjoint Operators ◽  
Usual Product

Let H be a Hilbert space with the usual product [x, y] and with an indefinite inner product (x, y) which, for some orthogonal decompositionin H, is defined bywhereand dim H1 = κ, a fixed positive integer.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

scholarly journals Local Spectral Properties Under Conjugations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Pietro Aiena ◽  
Fabio Burderi ◽  
Salvatore Triolo
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Analytic Core ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

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