By Carlos S. Kubrusly

By a Hilbert-space operator we suggest a bounded linear transformation be tween separable complicated Hilbert areas. Decompositions and versions for Hilbert-space operators were very energetic learn subject matters in operator concept over the last 3 many years. the most motivation in the back of them is the in variation subspace challenge: does each Hilbert-space operator have a nontrivial invariant subspace? this can be maybe the main celebrated open query in op erator conception. Its relevance is simple to provide an explanation for: common operators have invariant subspaces (witness: the Spectral Theorem), in addition to operators on finite dimensional Hilbert areas (witness: canonical Jordan form). If one concurs that every of those (i. e. the Spectral Theorem and canonical Jordan shape) is necessary adequate an success to push aside from now on justification, then the hunt for nontrivial invariant subspaces is a typical one; and a recalcitrant one at that. Subnormal operators have nontrivial invariant subspaces (extending the conventional branch), in addition to compact operators (extending the finite-dimensional branch), however the query continues to be unanswered even for both easy (i. e. basic to outline) specific sessions of Hilbert-space operators (examples: hyponormal and quasinilpotent operators). but the invariant subspace quest has under no circumstances been a failure in any respect, although faraway from being settled. the quest for nontrivial invariant subspaces has undoubtly yielded loads of great leads to operator concept, between them, these touching on decompositions and versions for Hilbert-space operators. This e-book comprises 9 chapters.

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**Extra resources for An Introduction to Models and Decompositions in Operator Theory**

**Example text**

An operator T E B[1-£] is similar to a contraction (similar to a strict contraction) if and only if there exists Q E g+[1-£] such that o ::: Q2 - T*Q2T (such that 0 -< Q2 - T*Q 2T). If T is a contraction, then we may take Q = I. ). Recall that 0 = 1- T*T if and only if T is an isometry (cf. 6). 8, then similarity to a contraction collapses to similarity to an isometry. 9. e. ). Proof. Let T be an operator on 'H.. ] such that T*Q2T = Q2. , because for all x E 1{. q. e. , /C] is unitary and IWI = (W*W)!

23,55 and 69]). C. S. 1. Let T be a contraction so that The operator A has the following properties. (a) o ::: A ::: I (b) IITnxl1 (c) TM AT n = A for every n ~ I (Le. A is a nonnegative contraction). I IIA"ixil as n = IIA'ixil I (d) IIA'irnxil (e) (1- A)rn ~ 0, I ~ 00 for all x E H. 1. )rn ~ IIATnxil (g) IIAII (h) AT =P 0 and T A =P 0 whenever A =P O. (i) N(A) U) N(1 - A) =1 = IIA'ixil as n ~ 00 1. o. (0 ~ ~ for all x E H. whenever A =P O. {x E = H: rnx ~ {x E OJ. H: IITnxil = IIxil 'In ~ I}.

An operator T on H is hyponormal if 0 ~ T*T - TT* (equivalently, if IIT*xll ~ IITxl1 for all x E H). Every subnormal operator is hyponormal. It is still unknown whether every hyponormal operator has a nontrivial invariant subspace. A result that comes very close to an afirmative answer says that any hyponormal operator whose spectrum has nonempty interior has a nontrivial invariant subspace [11] (also see [18]). In fact, since Putnam inequality [70] 22 An Introduction to Models and Decompositions in Operator Theory for hyponormal operators (JJT*T - TT*JJ :::: ]l'-l area(a(T») ensures that a hyponormal operator whose spectrum has zero area is normal, the cases not covered by the above two results comprise a class of operators with rather pathological spectra.