On similarity of operators to isometries (Q1803910)

Isometric and unitary operators being central to the theory of linear operators on Hilbert space, the problem of determining whether a given operator is similar to one of these assumes a natural importance and has been examined by various researchers since 1947, when Sz.-Nagy announced necessary and sufficient conditions for similarity to an isometry. An interesting and worthwhile contribution to this ongoing discussion is made in the paper under review which constitutes a portion of the author's Ph.D. Thesis written under the supervision of Professor Carl Pearcy at Texas A\(\&\)M University. (This reviewer continues to appreciate the several classes and seminars he took from Professor Pearcy while a student at The University of Michigan.) By way of preliminaries, if \(H\) is a (complex) Hilbert space with two inner products, \((\cdot,\cdot)\) and \((\cdot,\cdot)_ 0\), then the induced norms, \(\|\cdot\|\) and \(\|\cdot\|_ 0\), are equivalent if there are positive constants \(a\) and \(b\) such that \(a\| h\|\leq\| h\|_ 0\leq b\| h\|\) for all \(h\) in \(H\). In this case, the identity mapping \(\omega: (H,(\cdot,\cdot))\to (H,(\cdot,\cdot)_ 0)\) is an invertible bounded operator. For \(A\) a bounded linear operator on \(H\), let \(A_ 0=\omega A\omega^{-1}\). Also, if \(T\) is a contraction operator on \(H\) and \(U\), defined on a Hilbert space \(K\supset H\), is its minimal unitary dilation, then the *-residual subspace \({\mathcal R}_*\) of \(K\) is defined by \({\mathcal R}_*=\bigcap^ \infty_{k=0}\Bigl(\bigvee^ \infty_{n=k} U^{*n} H\Bigr)\). Several of the principal results of the paper answer the problem of similarity to an isometry in terms of the resolvent of the operator. For instance, a special case of Theorem 3.2 (see Remark 3.4) asserts that the surjective operator \(A\) is similar to an isometry if and only if there is an inner product \((\cdot,\cdot)_ 0\) on \(H\) such that (i) \(\|\cdot\|\sim\| \cdot\|_ 0\), (ii) \(A_ 0\) is power bounded (that is, \(\sup\{\| A^ n_ 0\|_ 0\}<\infty\)), and (iii) there exists \(c>0\) such that \(\|(A_ 0-\lambda)h\|_ 0\geq c(1-|\lambda|)\| h\|_ 0\) for all \(\lambda\) in the open unit disc and all \(h\) in \(H\). Another of the main theorems is Theorem 5.2 which gives a model for a contraction similar to an isometry. Specifically, it is shown that the contraction \(T\) is similar to an isometric operator if and only if the restriction of the projection \(P_{{\mathcal R}_*}\) to \(H\) is one-to-one and has closed range, in which case one has that \(T\) is similar to the restriction of its minimal unitary dilation \(U\) to the space \(P_{{\mathcal R}_*}H\).