Cauchy--Schwarz norm inequalities for weak*-integrals of operator valued functions
In this paper, \( \mathcal{H}\) is a separable, infinite-dimensional Hilbert space, \(\mathcal{B}( \mathcal{H})\) is the space of all bounded linear operators on \( \mathcal{H}\), and \( \mathcal{C}_{p} (\mathcal{H})\) (\(p \geq 1\)) are the Schatten \(p\)-ideals. Each symmetric gauge function \( \varPhi \) on sequences determines a symmetric norm on \(\mathcal{B}( \mathcal{H})\) defined by \( \| X \| _{ \varPhi } = \varPhi ( \{ s_{n}(X) \}_{n=1}^{ \infty})\) with \( s_{1}(X) \geq s_{2}(X) \geq \dots \) being the singular values of \(X\); the symbol \(| | | . | | | \) is used for any such norm on a naturally associated norm ideal \( \mathcal{C}_{ | | | . | | | } (\mathcal{H}) \) of \( \mathcal{C}_{ \infty} (\mathcal{H}) \). For a measure space \( (\Omega, \mathfrak{M}, \mu) \) and for a weak*-integrable mapping \( \mathcal{A}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}( \mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) (note that \(\mathcal{B}( \mathcal{H})\) is the dual space of \(\mathcal{C}_{1}( \mathcal{H})\)), for any \( E \in \mathfrak{M}\), one gets a unique \(\mathcal{I}_{E} \in \mathcal{B}( \mathcal{H})\) such that \[ \text{tr}( \mathcal{I}_{E} Y) = \int_{E}\text{tr}( \mathcal{A}_{t} Y) d \mu (t), \; \text{ for all }Y \in \mathcal{C}_{1}( \mathcal{H}). \] This is denoted by \( \int_{E} \mathcal{A}_{t} d \mu (t) \) or \( \int_{E} \mathcal{A} d \mu \). The main result is: Suppose that \((\Omega, \mathfrak{M}, \mu)\) is a \(\sigma\)-finite measure space and that \(\mathcal{A}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}(\mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) and \( \mathcal{B}: \Omega \to (\mathcal{B}( \mathcal{H}), \sigma(\mathcal{B}( \mathcal{H}), \mathcal{C}_{1}( \mathcal{H})))\) are \(\mu\)-weak* measurable. Then for every \(p, q, r \geq 1\) such that \( \frac{1}{q} + \frac{1}{r}=\frac{2}{p} \), and \(X \in \mathcal{C}_{p} (\mathcal{H})\), \[ \left\| \int_{\Omega} \mathcal{A} X \mathcal{B} \right\| _{p} \leq \left\| \sqrt{2q}{ \int_{\Omega} \mathcal{A^{*}} \left(\int_{\Omega} \mathcal{A} \mathcal{A^{*}} d \mu\right)^{q-1} \mathcal{A} d \mu} X \sqrt{2r}{ \int_{\Omega} \mathcal{B} \left(\int_{\Omega} \mathcal{B^{*}} \mathcal{B} d \mu\right)^{r-1} \mathcal{B^{*}} d \mu} \right\| _{p}. \] If both \( \{ \mathcal{A}_{t} \}_{t \in \Omega} \) and \(\{ \mathcal{B} \}_{t \in \Omega} \) consist of commuting normal operators, then for all unitary invariant norms \( | | | . | | | \) and all \( X \in \mathcal{C}_{ | | | . | | | } (\mathcal{H}) \), \[ \left| \left| \left| \int_{\Omega} \mathcal{A} X \mathcal{B} d \mu \right| \right| \right| \leq \left| \left| \left| \left(\sqrt{\int_{\Omega} \mathcal{A^{*}} \mathcal{A} d \mu}\right) X \left(\sqrt{\int_{\Omega} \mathcal{B^{*}} \mathcal{B} d \mu}\right) \right| \right| \right| . \] If, in addition, \(\int_{\Omega} \mathcal{A^{*}} \mathcal{A} d \mu \leq I\) and \(\int_{\Omega} \mathcal{B^{*}} \mathcal{B} d \mu \leq I\), then \[ \left(\sqrt{I- \int_{\Omega} \mathcal{A^{*}} \mathcal{A} d \mu}\right) X \left(\sqrt{I- \int_{\Omega} \mathcal{B^{*}} \mathcal{B} d \mu}\right)\in \mathcal{C}_{ | | | . | | | } (\mathcal{H}) \] and \[ \left| \left| \left| \left(\sqrt{ I- \int_{\Omega} \mathcal{A^{*}} \mathcal{A} d \mu}\right) X \left(\sqrt{ I- \int_{\Omega} \mathcal{B^{*}} \mathcal{B} d \mu}\right) \right| \right| \right| \leq \left| \left| \left| X- \int_{\Omega} \mathcal{A} X \mathcal{B} d \mu \right| \right| \right|. \] Several applications to some other operator inequalities and to operator monotone functions are given.
scientific article; zbMATH DE number 2136762