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Pusz-Woronowicz functional calculus and extended operator convex perspectives. (English) Zbl 07462024

F. Kubo and T. Ando [Math. Ann. 246, 205–224 (1980; Zbl 0412.47013)] gave a general theory of operator means and connections and proved that this theory has a close relation to Löwner’s theory on operator monotone functions. In fact, operator connections \(\sigma\) correspond one-to-one to non-negative operator monotone functions \(f\) on \([0, \infty)\) in such a way that \[ A \sigma B=A^{1/2}f\left(A^{-1/2}BA^{-1/2}\right)A^{1/2},\tag{1} \] where \(A\) and \(B\) are bounded positive operators on Hilbert space \(\mathcal{H}\), with \(A\) invertible. Also, the operator perspective associated with real continuous functions \(f\) on \((0, \infty)\) was introduced as \[ P_f(A, B)=B^{1/2}f\left(B^{-1/2}AB^{-1/2}\right)B^{1/2}, \] where \(A\) and \(B\) are invertible positive operators on a Hilbert space \(\mathcal{H}\). We know that \(P_f(A, B)\) is jointly operator convex iff \(f\) is operator convex on \((0, \infty)\).
The PW-functional calculus in the framework of operator theory is defined as follows:
Assume that \(A\) and \(B\) are positive operators on \(\mathcal{H}\) and the positive sesquilinear forms \(\alpha, \beta\) on \(\mathcal{H}\) give by \(\alpha(\xi, \eta):=\langle A\xi, \eta\rangle\) and \(\beta(\xi, \eta):=\langle B\xi, \eta\rangle\) for \(\xi, \eta\in\mathcal{H}\). Now, the PW-functional calculus \(\phi(\alpha, \beta)\) for \(A\) and \(B\) state that a bounded operator \(T_{A, B}:\mathcal{H}\longrightarrow\mathcal{H}_{A, B}:=\overline{\mathrm{ran}(A+B)}\) by \(T_{A, B}(\xi):=(A+B)^{1/2}\xi~~(\xi\in\mathcal{H})\), we have \(R_{A, B}, S_{A, B}\in \mathbb{B}(\mathcal{H}_{A, B})^+\) such that \[ R_{A, B}+S_{A, B}=I_{A, B}\,,\quad A=T_{A, B}^*R_{A, B}T_{A, B} \text{ and } B=T_{A, B}^*S_{A, B}T_{A, B}. \] Then \((T_{A, B}:\mathcal{H}\longrightarrow\mathcal{H}_{A, B}, R_{A, B}, S_{A, B})\) is compatible representation of \((\alpha, \beta)\) and \(\phi(\alpha, \beta)\) clearly consides with \[ \phi(A, B):=T_{A, B}^*\phi(R_{A, B}, S_{A, B})T_{A, B} \] if \(\phi\) is locally bounded and homogeneous Borel function on \([0, \infty)^2\). Obviously, Kubo-Ando’s operator connections \(A \sigma B=\phi(B, A)\) as PW-functional if \(\phi\) is the two-variable extension of representing function \(f\) in (1). Similarly, operator perspectives \(P_f(A, B)=\phi(A, B)\).
In this paper under review, the authors study in the framework of operator theory, Pusz and Woronowicz’s functional calculus for pairs operators in \(\mathbb{B}(\mathcal{H})\) associated with a homogeneous two-variable functions on \([0, \infty)^2\). Also, the authors give analyze convexity properties of the functional calculus. In continuation, the authors present the very nice results of this concept.

MSC:

47A60 Functional calculus for linear operators
47A64 Operator means involving linear operators, shorted linear operators, etc.
47A63 Linear operator inequalities
47B65 Positive linear operators and order-bounded operators
47A07 Forms (bilinear, sesquilinear, multilinear)

Citations:

Zbl 0412.47013
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Full Text: DOI arXiv

References:

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