Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range

2003 ◽  
Vol 46 (2) ◽  
pp. 216-228 ◽  
Author(s):  
Chi-Kwong Li ◽  
Leiba Rodman ◽  
Peter Šemrl
Keyword(s):  
Linear Space ◽  
Numerical Range ◽  
Positive Definiteness ◽  
Linear Operators ◽  
Bounded Linear ◽  
Real Linear Space ◽  
The Real ◽  
Selfadjoint Operators ◽  
Linear Maps ◽  
Real Linear

AbstractLet H be a complex Hilbert space, and be the real linear space of bounded selfadjoint operators on H. We study linear maps ϕ: → leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps ϕ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption ϕ has the form , for a suitable invertible or unitary T and ξ ∈ {1, −1}, where Xt stands for the transpose of X relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on H. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form αI + K, where α is a scalar and K is compact.

On the Hahn-Banach Extension Property

1970 ◽  
Vol 13 (1) ◽  
pp. 9-13
Author(s):  
Ting-On To
Keyword(s):  
Linear Function ◽  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Real Linear

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

scholarly journals ON EXTREME POINTS OF A CERTAIN LINEAR SPACE OF LOCALLY UNIVALENT FUNCTIONS

1992 ◽  
Vol 23 (4) ◽  
pp. 321-325
Author(s):  
KUALIDA INAYAT NOOR
Keyword(s):  
Linear Space ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Space Structure ◽  
Real Linear Space ◽  
The Real ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Real Linear

Let $H = (H, \oplus, \odot)$ denote the real linear space of locally univalent normalized functions in the unit disc as defined by Hornich. For $-1\le B <A\le 1$, $k>2$, the classes $V_k[A,B]$ of functions with bounded boundary rotation are introduced and this linear space structure is used to determine the extreme points of the classes $V_k[A,B]$.

scholarly journals Nearly directed subspaces of partially ordered linear spaces

1968 ◽  
Vol 16 (2) ◽  
pp. 135-144
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Transitive Relation ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Partially Ordered ◽  
Real Linear ◽  
Image Position ◽  
Ordered Linear Spaces

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

Linear Normed Spaces with Extension Property

1966 ◽  
Vol 9 (4) ◽  
pp. 433-441 ◽  
Author(s):  
George Elliott ◽  
Israel Halperin
Keyword(s):  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Linear Mapping ◽  
Normed Spaces ◽  
Bounded Linear ◽  
The Real ◽  
Real Normed Linear Space

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

About the Polarity in a Real Linear Space

1977 ◽  
Vol 77 (1) ◽  
pp. 181-185 ◽  
Author(s):  
Jacques Bair
Keyword(s):  
Linear Space ◽  
Real Linear Space ◽  
Real Linear

Some reiteration results for interpolation methods defined by means of polygons

2008 ◽  
Vol 138 (6) ◽  
pp. 1179-1195 ◽  
Author(s):  
Fernando Cobos ◽  
Luz M. Fernández-Cabrera ◽  
Joaquim Martín
Keyword(s):  
Banach Algebras ◽  
Function Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Real ◽  
Interpolation Methods ◽  
Real Method

We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.

Complex strict and uniform convexity and hyponormal operators

1984 ◽  
Vol 96 (3) ◽  
pp. 483-493 ◽  
Author(s):  
Kirsti Mattila
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Uniform Convexity ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Image Position

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the setIf V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

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