Equivalence

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University — PUC/RJ

Chapter Chapter 1 in An Introduction to Models and Decompositions in Operator Theory, 1997, pp 23-35 from Springer

Abstract: Abstract Two Banach spaces are topologically isomorphic if there exists a linear home- omorphism between them. Two Hilbert spaces, say H and K, are topologically isomorphic if and only ifthey are unitarily equivalent. Indeed, if W ∈ G[H, K], set ∣W∣ = (W*W)1/2 ∈ G+[H] and note that U: = W∣W∣−l ∈ G[H, K] is unitary (i.e. U*U = I on H and U U* = I on K). Actually U ∣W∣ is the polar decomposition of W (cf. Remark 0.10). Therefore $$ G\left[ {H, K} \right] \ne \not{0} \Leftrightarrow \{ U \in G[H, K]:U is unitary\} \ne \not{0} $$ .

Date: 1997
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DOI: 10.1007/978-1-4612-1998-9_2

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