scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

Range Inclusion for Multilinear Mappings: Applications

1985 ◽  
Vol 28 (3) ◽  
pp. 317-320
Author(s):  
C. K. Fong
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Trace Class ◽  
Inner Derivation ◽  
Finite Rank Operators ◽  
Multilinear Mappings ◽  
Trace Class Operators ◽  
The Ideal

AbstractThe result of S. Grabiner [5] on range inclusion is applied for establishing the following two theorems: 1. For A, B ∊ L(H), two operators on the Hilbert space H, we have DBC0(H) ⊆ DAL(H) if and only if DBC1(H) ⊆ DAL(H), where DA is the inner derivation which sends S ∊ L(H) to AS - SA, C1(H) is the ideal of trace class operators and C0(H) is the ideal of finite rank operators. 2. (Due to Fialkow [3]) For A, B ∊ L(H), we write T(A, B) for the map on L(H) sending S to AS - SB. Then the range of T(A, B)is the whole L(H) if it includes all finite rank operators L(H).

scholarly journals On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 903
Author(s):  
Marat V. Markin ◽  
Edward S. Sichel
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Symmetric Operators ◽  
Straightforward Proof

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set𝒮r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the𝒮r(ℬ)setting.

COVARIANCE-PARAMETER LÉVY PROCESSES IN THE SPACE OF TRACE-CLASS OPERATORS

2005 ◽  
Vol 08 (01) ◽  
pp. 33-54 ◽  
Author(s):  
VÍCTOR PÉREZ-ABREU ◽  
ALFONSO ROCHA-ARTEAGA
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Lévy Processes ◽  
Trace Class ◽  
Levy Processes ◽  
Trace Class Operators

The paper deals with Lévy processes with values in L1(H), the Banach space of trace-class operators in a Hilbert space H. Lévy processes with values and parameter in a cone K of L1(H) are introduced and several properties are established. A family of L1(H)-valued Lévy processes is obtained via the subordination of K-parameter, L1(H)-valued Lévy processes, identifying explicitly their generating triplets.

scholarly journals On Classical Representations of Convex Descriptions

1993 ◽  
Vol 48 (3) ◽  
pp. 469-470 ◽  
Author(s):  
Slawomir Bugajski
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Vector Lattice ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Classical Representation ◽  
Base Norm

Abstract It is demonstrated that if V* is not a vector lattice, where V is a base norm Banach space, then there is no commutative observable providing a classical representation for V. This observation generalizes a similar result of Busch and Lahti, obtained for V - the trace class of operators on a separable complex Hilbert space.

PROBABILITY STRUCTURES FOR QUANTUM STATE SPACES

1995 ◽  
Vol 07 (07) ◽  
pp. 1105-1121 ◽  
Author(s):  
PAUL BUSCH ◽  
GIANNI CASSINELLI ◽  
PEKKA J. LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Quantum State ◽  
Positive Operator ◽  
Trace Class ◽  
Measurable Space ◽  
Probability Measures ◽  
State Spaces ◽  
Trace Class Operators ◽  
Positive Operator Valued Measure

The theme of this paper is to represent the states of a quantum system by means of probability measures. We fix a positive operator valued measure E on a measurable space (Ω, ℬ(Ω)) acting in a Hilbert space ℋ, and we study the properties of the mapping that it induces from the set of trace class operators on ℋ to the set of measures on (Ω, ℬ(Ω)). In particular, the injectivity and the surjectivity of this map are characterised in terms of the properties of E.

scholarly journals Spectral radius inequalities for functions of operators defined by power series

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2847-2856
Author(s):  
S.S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Linear Operator ◽  
Spectral Radius ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Commuting Operators ◽  
Functions Of Operators ◽  
The Absolute

By the help of power series f(z)=??,n=0 anzn we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f , namely fa(z):= ??,n=0 |an|zn. Utilising these functions we show among others that r[f(T)] ? fa [r(T)] where r (T) denotes the spectral radius of the bounded linear operator T on a complex Hilbert space while ||T|| is its norm. When we have A and B two commuting operators, then r2[f(AB)]? fa(r2(A)) fa(r2(B)) and r[f(AB)]?1/2[fa(||AB||)+fa(||A2||1/2||B2||1/2)].

scholarly journals Remarks on n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5441-5451 ◽  
Author(s):  
Muneo Chō ◽  
Ji Lee ◽  
Kotaro Tanahashi ◽  
Atsushi Uchiyama
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Normal Operators

Let T be a bounded linear operator on a complex Hilbert space and n,m ? N. Then T is said to be n-normal if T+Tn = TnT+ and (n,m)-normal if T+mTn = TnT+m. In this paper, we study several properties of n-normal, (n,m)-normal operators. In particular, we prove that if T is 2-normal with ?(T) ? (-?(T)) ? {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n,m)-normal, then T is decomposable and Weyl?s theorem holds for f (T), where f is an analytic function on ?(T) which is not constant on each of the components of its domain.

scholarly journals A Counterexample in the theory of derivations

1989 ◽  
Vol 31 (2) ◽  
pp. 161-163
Author(s):  
Feng Wenying ◽  
Ji Guoxing
Keyword(s):  
Hilbert Space ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Schmidt Norm ◽  
Schmidt Operator ◽  
Image Position

Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H. Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H). It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace). For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]). In [1], Fuad Kittaneh proved the following result.

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