Singular value inequalities involving convex and concave functions of positive semidefinite matrices. (English) Zbl 1446.15004

Alrimawi, Fadi; Hirzallah, Omar; Kittaneh, Fuad

Ann. Funct. Anal. 11, No. 4, 1257-1273 (2020).

Summary: Let \(A\) and \(B\) be \(n\times n\) positive semidefinite matrices, and let \(\alpha,\beta\in (0,1)\) such that \(\alpha+\beta=1\). Among other inequalities, it is shown that

(a)

If \(f\) is a non-negative concave function on \([0,\infty)\), then \[ s_j(\alpha f(A)+\beta f(B))\leq s_j(f(\sqrt{2}\left|\alpha A+i\beta B\right|)) \] for \(j=1,\dots,n\).

(b)

If \(f\) is a non-negative strictly increasing convex function on \([0,\infty)\) with \(f(0)=0\), then \[ s_j(f\left(\alpha A+\beta B\right))\leq\sqrt{2}s_j(\alpha f\left(A\right) +i\beta f\left(B\right)) \] for \(j=1,\dots ,n\). Here \(s_j\left(X\right)\) denotes the largest \(j\)th singular value of the matrix \(X\).

MSC:

15A18	Eigenvalues, singular values, and eigenvectors
15A42	Inequalities involving eigenvalues and eigenvectors
15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

accretive-dissipative matrix; positive semidefinite matrix; singular value; unitarily invariant norm; convex function; concave function; inequality

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