An Introduction to Models and Decompositions in Operator by Carlos S. Kubrusly PDF

By Carlos S. Kubrusly

ISBN-10: 1461219981

ISBN-13: 9781461219989

ISBN-10: 1461273749

ISBN-13: 9781461273745

By a Hilbert-space operator we suggest a bounded linear transformation be­ tween separable complicated Hilbert areas. Decompositions and types for Hilbert-space operators were very lively learn themes in operator idea over the last 3 a long time. the most motivation at the back of them is the in­ version subspace challenge: does each Hilbert-space operator have a nontrivial invariant subspace? this can be possibly the main celebrated open query in op­ erator idea. Its relevance is simple to provide an explanation for: general 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 brush aside any more justification, then the hunt for nontrivial invariant subspaces is a usual 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 basic (i. e. basic to outline) specific sessions of Hilbert-space operators (examples: hyponormal and quasinilpotent operators). but the invariant subspace quest has by no means 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 types for Hilbert-space operators. This e-book comprises 9 chapters.

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Example text

Sy is a unilateral shift of mUltiplicity y = Lj,:-J Pj if and only if Sy = EBj,:-J Sf3J where Sf3J is a unilateral shift of multiplicity Pj for each j). 4 is a particular case of it where Pj = P for every j. 4: instead of taking a fixed p-dimensional Hilbert space 'H, take a (countable)collectionofa(separable)Hilbertspaces'Hj,{'Hj ; 0 ~ j ~ a-I}, such that dim('Hj) = Pj for each j. e~ ('H) replaced by EB~o 'Hj for each j. A unilateral shift of finite multiplicity, say m, may also be defined as the mth power of a unilateral shift of mUltiplicity 1.

Shifts This shows at once that and also that s S~S+ =I (equivalently, s~n S~ =I for every n 2: 1), O. Indeed, for each n 2: 1, s~n --* EB Uk+ 00 s~nx = 1 ••• Uk+nXk+n k=O so that Ils~nxl12 = L~o IIU;+l ... U;+nxk+nI12 = L~n Ilxklf ~ 0 as ~ 00. Therefore a unilateral shift is an isometry and its adjoint (which is called a backward unilateral shift) is strongly stable. C o ~ 1io be unitary so that dim(1ik) = dim (/Co) for every k 2: O. Co). Co. Co). This is enough to ensure the following result. 2.

An operator is a unilateral shift of multiplicity mp if and only if it is the mth power of a unilateral shift of multiplicity p. Proof. Let S+ be a unilateral shift of multiplicity Pacting on a Hilbert space 'H, and let {'Hk; k ::: O} be the underlying orthogonal sequence of p-dimensional subspaces of'H = EB~o 'Hk. Take an arbitrary positive integer m and set (k+l)m-l Mk = EB 'Hj j=km for each k ::: O. Note that {Mk; k ::: O} is a sequence of mp-dimensional orthogonal subspaces of'H such that 'H = EB~ M k • Moreover (k+l)m-l (k+l)m (k+l)m S+(Mk) = S+( EB'Hj) = EBUj('Hj-l) = EB'Hj, j=km j=km+l j=km+l so that a trivial induction leads to (k+l)m+m-l = EB'Hj j=km+m (k+2)m-l = EB'Hj j=(k+l)m for every k ::: o.

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An Introduction to Models and Decompositions in Operator Theory by Carlos S. Kubrusly

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