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.

ON ∞-COMPLEX SYMMETRIC OPERATORS

2017 ◽  
Vol 60 (1) ◽  
pp. 35-50
Author(s):  
MUNEO CHŌ ◽  
EUNGIL KO ◽  
JI EUN LEE
Keyword(s):  
Spectral Properties ◽  
Symmetric Operator ◽  
Complex Symmetric Operator ◽  
Decomposition Property ◽  
Complex Symmetric ◽  
Symmetric Operators

AbstractIn this paper, we study spectral properties and local spectral properties of ∞-complex symmetric operators T. In particular, we prove that if T is an ∞-complex symmetric operator, then T has the decomposition property (δ) if and only if T is decomposable. Moreover, we show that if T and S are ∞-complex symmetric operators, then so is T ⊗ S.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

scholarly journals On local spectral properties of complex symmetric operators

2011 ◽  
Vol 379 (1) ◽  
pp. 325-333 ◽  
Author(s):  
S. Jung ◽  
E. Ko ◽  
M. Lee ◽  
J. Lee
Keyword(s):  
Spectral Properties ◽  
Complex Symmetric ◽  
Symmetric Operators

scholarly journals Some notes on complex symmetric operators

2021 ◽  
Vol 2021 (1) ◽  
pp. 90-96
Author(s):  
Marcos S. Ferreira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Bounded Operator ◽  
Complex Symmetric ◽  
Symmetric Operators

Abstract In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * 𝒥T, where T is an unitary operator and 𝒥 f ( z ) = f ( z ¯ ) ¯ \mathcal{J}f\left( z \right) = \overline {f\left( {\bar z} \right)} with f ∈ H 2. Moreover we prove some relations of complex symmetry between the operators T and |T|, where T = U |T| is the polar decomposition of bounded operator T ∈ ℒ(ℋ) on the separable Hilbert space ℋ.

scholarly journals Weyl type theorems of 2×2 upper triangular operator matrices

2016 ◽  
Vol 434 (2) ◽  
pp. 1065-1076
Author(s):  
Qingmei Bai ◽  
Junjie Huang ◽  
Alatancang Chen
Keyword(s):  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices

scholarly journals On symmetric and skew-symmetric operators

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 293-303 ◽  
Author(s):  
Chafiq Benhida ◽  
Muneo Chō ◽  
Eungil Ko ◽  
Ji Lee
Keyword(s):  
Spectral Properties ◽  
Complex Symmetric ◽  
Symmetric Operators

In this paper we show many spectral properties that are inherited by m-complex symmetric and m-skew complex symmetric operators and give new results or recapture some known ones for complex symmetric operators.

scholarly journals On upper triangular operator 2 x 2 matrices over C*-algebras

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 691-706
Author(s):  
Stefan Ivkovic
Keyword(s):  
Operator Matrices ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
C Algebra ◽  
C Algebras ◽  
Direct Sum ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Definition Of ◽  
The Relationship

We study adjointable, bounded operators on the direct sum of two copies of the standard Hilbert C*-module over a unital C*-algebra A that are given by upper triangular 2 by 2 operator matrices. Using the definition of A-Fredholm and semi-A-Fredholm operators given in [3], [4], we obtain conditions relating semi-A-Fredholmness of these operators and that of their diagonal entries, thus generalizing the results in [1], [2]. Moreover, we generalize the notion of the spectra of operators by replacing scalars by the elements in the C*-algebra A: Considering these new spectra in A of bounded, adjointable operators on Hilbert C*-modules over A related to the classes of A-Fredholm and semi-A-Fredholm operators, we prove an analogue or a generalized version of the results in [1] concerning the relationship between the spectra of 2 by 2 upper triangular operator matrices and the spectra of their diagonal entries.

Complex symmetric operators and semiclassical resonances

2018 ◽  
Vol 11 (03) ◽  
pp. 1850044
Author(s):  
Bekkai Messirdi ◽  
Sofiane Messirdi ◽  
Miloud Messirdi
Keyword(s):  
Hilbert Space ◽  
Complex Scaling ◽  
Quantum Resonances ◽  
Complex Symmetric ◽  
Symmetric Operators

Unbounded complex symmetric operators on Hilbert space are discussed and various spectral results on these operators are presented in this paper. In addition, we show that the technique of complex scaling allows one to identify the quantum resonances of complex self-adjoint Schrödinger Hamiltonians.

Weyl type theorems for complex symmetric operator matrices

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2891-2900 ◽  
Author(s):  
Il An ◽  
Eungil Ko ◽  
Ji Lee
Keyword(s):  
Symmetric Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator Matrices ◽  
Complex Symmetric Operator ◽  
Weyl Type Theorems ◽  
Complex Symmetric ◽  
Necessary And Sufficient

In this paper, we study Weyl type theorems for complex symmetric operator matrices. In particular, we give a necessary and sufficient condition for complex symmetric operator matrices to satisfy a-Weyl?s theorem. Moreover, we also provide the conditions for such operator matrices to satisfy generalized a-Weyl?s theorem and generalized a-Browder?s theorem, respectively. As some applications, we give various examples of such operator matrices which satisfy Weyl type theorems.

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