Inequalities for unitarily invariant norms (Q2902021)

Let \(\|.\|\) denote any unitarily invariant norm on the space \(M_n\) of \(n\times n\) complex matrices, that is, \(\|.\|\) satisfies \(\| UAV\|=\| A\|\) for any \(A\) in \(M_n\) and any unitary \(U\) and \(V\) in \(M_n\). It is known that such a norm is a symmetric gauge function of the singular values of a matrix and they include Ky Fan \(k\)-norms \((1\leq k\leq n)\) and Schatten \(p\)-norms \((1\leq p<\infty)\).NEWLINENEWLINE In this paper, the authors prove several inequalities for unitarily invariant norms. For example, in Section 2, they prove two inequalities for the Hilbert-Schmidt norm \(\|.\|_2\) (or the Schatten \(2\)-norm), which give upper bounds for \(\| AX+ XB\|_2\), where \(A\), \(B\) and \(X\) are all in \(M_n\) and \(A\) and \(B\) are positive semidefinite. In Section 3, two new inequalities for unitarily invariant norms \(\|.\|\) are obtained. For example, it is shown that \(\| A(A^*A+ B^*B)B^*\|\leq \|(AA^*+ BB^*)^2\|/2\) for any \(A\) and \(B\) in \(M_n\). When \(A\) and \(B\) are positive semidefinite, this yields an inequality proven by Bhatia and Kittaneh before. Section 4 gives a Hölder-type inequality for which generalizes the ones by Kittaneh and Manasrah and by Horn and Zhan, namely, if \(A\), \(B\) and \(X\) are in \(M_n\) with \(A\) and \(B\) positive semidefinite and if \(p\), \(q\), \(r>0\) with \(1/p+ 1/q= 1\), then NEWLINE\[NEWLINE\|\,|AXB|^r\|+{1\over s}(\|\,|A^p X|^r\|^{1/2}- \|\,|XB^q|^r\|^{1/2})^2\leq {1\over p}\|\,|A^pX|^r\|+ {1\over q}\|\,|XB^q|^r\|,NEWLINE\]NEWLINE where \(s=\max\{p,q\}\). Other inequalities involving four matrices in \(M_n\) are also obtained.