scholarly journals Positive linear maps on Hilbert space operators and noncommutative Lp spaces

2021 ◽  
Vol 87 (12) ◽  
pp. 195-206
Author(s):  
Jean-Christophe Bourin ◽  
Jingjing Shao
Keyword(s):  
Hilbert Space ◽  
Lp Spaces ◽  
Hilbert Space Operators ◽  
Linear Maps ◽  
Positive Linear Maps

scholarly journals Improved Kantorovich and Wielandt operator inequalities for positive linear maps

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 871-876 ◽  
Author(s):  
Wenshi Liao ◽  
Junliang Wu
Keyword(s):  
Hilbert Space ◽  
Operator Inequalities ◽  
Hilbert Space Operators ◽  
Linear Maps ◽  
Positive Linear Maps

This paper improves and generalizes the Kantorovich and Wielandt inequalities for positive linear maps on Hilbert space operators and presents more general and precise results compared to many recent results.

Semipositivity of linear maps relative to proper cones in finite dimensional real Hilbert spaces

2018 ◽  
Vol 34 ◽  
pp. 304-319 ◽  
Author(s):  
Chandrashekaran Arumugasamy ◽  
Sachindranath Jayaraman ◽  
Vatsalkumar Mer
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Hilbert Space ◽  
Linear Map ◽  
Proper Cone ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Lorentz Cone ◽  
Positive Linear Maps ◽  
Lyapunov’S Theorem

For a proper cone $K$ in a finite dimensional real Hilbert space $V$, a linear map $L$ is said to be $K$-semipositive if there exists $d \in K^\circ$, the interior of $K$, such that $L(d) \in K^\circ$. The aim of this manuscript is to characterize $K$-semipositivity of linear maps relative to a proper cone. Among several results obtained, $K$-semipositivity is characterized in terms of products of the form $YX^{-1}$ for $K$-positive linear maps ($L(K \setminus \{0\}) \subseteq K^\circ$) with $X$ invertible, semipositivity of matrices relative to the $n$-dimensional Lorentz cone $\mathcal{L}^n_{+}$ is characterized, semipositivity of the following three linear maps relative to the cone $\mathcal{S}^n_{+}$: $X \mapsto AXB$ (denoted by $M_{A,B}$), $X \mapsto AXB + B^tXA^t$ (denoted by $L_{A,B}$), where $A, B \in M_n(\reals)$, and $X \mapsto X - AXA^t$ (denoted by $S_A$, known as the Stein transformation) is characterized. It is also proved that $M_{A,B}$ is semipositive if and only if $B = \alpha A^t$ for some $\alpha > 0$, the map $L_{A,B}$ is semipositive if and only if $A(B^t)^{-1}$ is positive stable. A particular case of the new result generalizes Lyapunov's theorem. Decompositions of the above maps (when they are semipositive) in the form $L_1L_2^{-1}$, where $L_1$ and $L_2$ are both positive and invertible (assuming $A$ is invertible in the case of $S_A$) are presented. Moreover, a question on invariance of the semipositive cone $\mathcal{K}_A$ of a matrix under $A$ is partially answered.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

scholarly journals Bounded point evaluations for cyclic Hilbert space operators

2003 ◽  
Vol 4 (2) ◽  
pp. 301
Author(s):  
A. Bourhim
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations

<p>In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.</p>

scholarly journals Eigenvalue inequalities for differences of means of Hilbert space operators

2012 ◽  
Vol 436 (5) ◽  
pp. 1516-1527 ◽  
Author(s):  
Omar Hirzallah ◽  
Fuad Kittaneh ◽  
Mario Krnić ◽  
Neda Lovričević ◽  
Josip Pečarić
Keyword(s):  
Hilbert Space ◽  
Hilbert Space Operators
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