Some Convexity Features Associated with Unitary Orbits

2003 ◽  
Vol 55 (1) ◽  
pp. 91-111 ◽  
Author(s):  
Man-Duen Choi ◽  
Chi-Kwong Li ◽  
Yiu-Tung Poon
Keyword(s):  
Convex Hull ◽  
Linear Space ◽  
Spectral Properties ◽  
Numerical Range ◽  
Symmetric Matrices ◽  
Hermitian Matrices ◽  
Linear Transformations ◽  
Unitary Similarity ◽  
Similarity Orbit ◽  
Real Linear

AbstractLet be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

The numerical range of functions and best approximation

1974 ◽  
Vol 76 (1) ◽  
pp. 133-141 ◽  
Author(s):  
Lawrence A. Harris
Keyword(s):  
Convex Hull ◽  
Linear Space ◽  
Best Approximation ◽  
Normed Linear Space ◽  
Numerical Range ◽  
Closed Convex Hull ◽  
Compact Spaces ◽  
Range Of Functions ◽  
Function Mapping

In this note, we state general conditions which imply that the numerical range of a function mapping a set S into a normed linear space Y is the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

Pencils of hermitian matrices

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Symmetric Matrices ◽  
Canonical Forms ◽  
Hermitian Matrices ◽  
Analogous Statement ◽  
Real Linear ◽  
Simultaneous Diagonalizability

This chapter is concerned with the case when both matrices A and B are hermitian. Full and detailed proofs of the canonical forms under strict equivalence and simultaneous congruence are provided, based on the Kronecker form of the pencil A + tB. Several variations of the canonical forms are included as well. Among applications here are: the criteria for existence of a nontrivial positive semidefinite real linear combination and sufficient conditions for simultaneous diagonalizability of two hermitian matrices under simultaneous congruence. A comparison is made with pencils of real symmetric or complex hermitian matrices. It turns out that two pencils of real symmetric matrices are simultaneously congruent over the reals if and only if they are simultaneously congruent over the quaternions. An analogous statement holds true for two pencils of complex hermitian matrices.

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.

scholarly journals The boundary of the numerical range

1981 ◽  
Vol 22 (1) ◽  
pp. 69-72 ◽  
Author(s):  
G. de Barra
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Similar Condition ◽  
Point Spectrum ◽  
Numerical Range ◽  
Normal Operator ◽  
Density Condition ◽  
Compact Normal Operator

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

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 Inconsistency evaluation in pairwise comparison using norm-based distances

2020 ◽  
Vol 43 (2) ◽  
pp. 657-672 ◽  
Author(s):  
Michele Fedrizzi ◽  
Nino Civolani ◽  
Andrew Critch
Keyword(s):  
Linear Space ◽  
General Framework ◽  
Pairwise Comparison ◽  
Equivalence Classes ◽  
Preference Aggregation ◽  
Symmetric Matrices ◽  
Pairwise Comparison Matrix ◽  
Group Preference ◽  
Common Value ◽  
Inconsistency Index

AbstractThis paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation.

scholarly journals On partial isometries with circular numerical range

2021 ◽  
Vol 8 (1) ◽  
pp. 176-186
Author(s):  
Elias Wegert ◽  
Ilya Spitkovsky
Keyword(s):  
Blaschke Product ◽  
Partial Isometry ◽  
Shift Operator ◽  
Numerical Range ◽  
Elliptic Functions ◽  
Blaschke Products ◽  
Partial Isometries ◽  
Unitary Similarity ◽  
Boundary Interpolation ◽  
Poncelet Polygons

Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

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