Interpolation problems for operators with corank in \(\operatorname{Alg}\mathcal L\) (Q2871212)

A problem of longstanding interest in the theory of operators on Hilbert space is to find conditions under which, for given operators \(X\) and \(Y\), there is an operator \(A\) for which \(AX=Y\). Usually, we want \(A\) to also satisfy some additional requirements. In the paper under review, the author tackles a version of this classical interpolation problem.NEWLINENEWLINESpecifically, let \({\mathcal H}\) be a separable Hilbert space, let \({\mathcal L}\) be a lattice of subspaces of \({\mathcal H}\), with \(\operatorname{Alg}{\mathcal L}\) denoting the algebra of all bounded linear operators on \({\mathcal H}\) that leave invariant every subspace in \({\mathcal L}\), and let \(X\) and \(Y\) be given bounded linear operators on \({\mathcal H}\). In Theorem 2.1, whose proof relies on a classical interpolation result due to \textit{R. G. Douglas} [Proc. Am. Math. Soc. 17, 413--415 (1966; Zbl 0146.12503)], the author shows that, if there is a vector \(h\in{\mathcal H}\) such that \({\mathcal H}\) is the direct sum of the closure of the range of \(X\) and the one-dimensional subspace spanned by \(h\) and such that \(\langle h,E^\perp Xf\rangle=0\) for all \(E\in{\mathcal L}\) and for all \(f\in{\mathcal H}\), then there exists an operator \(A\in \operatorname{Alg}{\mathcal L}\) satisfying \(AX=Y\) if and only if NEWLINE\[NEWLINE\sup \left\{ {{\|E^\perp Yf\|}\over{\|E^\perp Xf\|}}:f\in{\mathcal H},\;E\in{\mathcal L}\right\}=K<\infty.NEWLINE\]NEWLINE Moreover, if this condition holds, then \(A\) can be chosen so that \(\|A\|=K\).NEWLINENEWLINEThe author then generalizes this result over several increments to arrive at Theorem 2.11, which gives a necessary and sufficient condition for the existence of an operator \(A\in \operatorname{Alg}{\mathcal L}\) that simultaneously satisfies \(AX_i=Y_i\) for countable collections of operators \(\{ X_i\}, \{Y_i\}\), where the hypothesis on \({\mathcal H}\) is that it is the direct sum of the closure of the range of one of the \(X_k\) and the subspace spanned by an orthonormal set of vectors \(\{h_j\}\) for which \(\langle h_j,E^\perp X_if\rangle\) for all \(i\), all \(j\), all \(E\in{\mathcal L}\), and all \(f\in{\mathcal H}\).NEWLINENEWLINESection 3 of the paper looks at the equation \(Ax=y\), for given vectors \(x, y\), by applying Theorem 2.1 to the rank-one operators \(X=x\otimes g^*\) and \(Y=y\otimes g^*\). (That is, \(Xf=\langle f,g\rangle x\) and \(Yf=\langle f,g\rangle y\). The initial theorem, Theorem 3.1, is again generalized to systems of countably many equations.NEWLINENEWLINEThe exposition suffers a bit from repetitiveness (for instance, \({\mathcal L}\) and \(\operatorname{Alg}{\mathcal L}\) are defined no fewer than three times) as well as from numerous small grammatical lapses. Nonetheless, the paper is quite readable and the results are interesting.