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An expression for the subdifferential set of the joint numerical radius of a tuple of matrices is obtained. Its applications to the Birkhoff-James orthogonality in the space consisting of tuples of matrices equipped with the joint numerical radius norm are discussed. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

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Mathematics

(SoNS) School of Natural Sciences

Linear and Multilinear Algebra

Subdifferential set of the joint numerical radius of a tuple of matrices

The matrix (formula presented), has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of S in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition. We give a bidiagonal decomposition of (formula presented) for positive integers (formula presented). We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices. © 2022, Element D.O.O.. All rights reserved.

Operators and Matrices

BIDIAGONAL DECOMPOSITIONS AND TOTAL POSITIVITY OF SOME SPECIAL MATRICES

For a unital C*-algebra A and a subspace B of A, a characterization for a best approximation to an element of A in B is obtained. As an application, a formula for the distance of an element of A from B has been obtained, when a best approximation of that element to B exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert C*-module to a subspace is obtained. © 2021 Ramanujan Mathematical Society. All rights reserved.

Journal of the Ramanujan Mathematical Society

Best approximations, distance formulas and orthogonality in C**-algebras

In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained. © 2021, Springer Nature Switzerland AG.

Operator Theory: Advances and Applications

Birkhoff–James Orthogonality and Applications: A Survey

It is shown that for positive real numbers 0&lt;λ1&lt;…&lt;λn, [Formula presented], where β(⋅,⋅) denotes the beta function, is infinitely divisible and totally positive. For [Formula presented], the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i+j)] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive. © 2020 Elsevier Inc.

Fulltext

Linear Algebra and Its Applications

Positivity properties of some special matrices

We obtain necessary and sufficient conditions for a matrix A to be Birkhoff-James orthogonal to another matrix B in the Ky Fan k-norms. A characterization for A to be Birkhoff-James orthogonal to any subspace W of M(n) is also obtained. © 2016 Informa UK Limited, trading as Taylor & Francis Group.

Orthogonality of matrices in the Ky Fan k-norms

We obtain a necessary and sufficient condition for a matrix A to be Birkhoff-James orthogonal to any subspace W of Mn(C). Using this we obtain an expression for the distance of A from any unital C*- subalgebra of Mn(C). © 2013 Elsevier Inc. All rights reserved.

Orthogonality to matrix subspaces, and a distance formula

The Birkhoff-James orthogonality is a generalization of Hilbert space orthogonality to Banach spaces. We investigate this notion of orthogonality when the Banach space has more structures. We start by doing so for the Banach space of square matrices moving gradually to all bounded operators on any Hilbert space, then to an arbitrary C*-algebra and finally a Hilbert C*-module. © 2013 Elsevier Ltd.

Journal of Mathematical Analysis and Applications

Characterization of Birkhoff-James orthogonality

The norm of the mth derivative of the map that takes an operator to its fcth antisymmetric tensor power is evaluated. The case m = 1 has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland. Variation of Grassman powers and spectra. Linear Algebra and its Applications, 40:1-18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive m-linear map between C*-algebras attains its norm at the m-tuple (I,I,...,I). Expressions for derivatives of the maps that take an operator to its fcth tensor power and fcth symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated.