New PDF release: Composition Operators on Function Spaces

By R.K. Singh and J.S. Manhas (Eds.)

ISBN-10: 0444815937

ISBN-13: 9780444815934

This quantity of the "Mathematics stories" sequence provides paintings performed on composition operators over the last 25 years. Composition operators shape an easy yet fascinating category of operators having interactions with varied branches of arithmetic and mathematical physics. After an advent, the e-book offers with those operators on Lp-spaces. This research turns out to be useful in measurable dynamics, ergodic idea, classical mechanics and Markov approach. The composition operators on sensible Banach areas (including Hardy areas) are studied in bankruptcy III. This bankruptcy makes touch with the speculation of analytic capabilities of complicated variables. bankruptcy IV offers a research of those operators on in the neighborhood convex areas of continuing services making touch with topological dynamics. within the final bankruptcy of the publication a few functions of composition operators in isometries, ergodic idea and dynamical structures are provided. an engaging interaction of algebra, topology, and research is displayed. This accomplished and updated research of composition operators on various functionality areas may still entice learn employees in practical research and operator concept, postgraduate scholars of arithmetic and information, in addition to to physicists and engineers.

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Let (X, Y , m ) be a finite atomic measure space and let CT be a composition operator on L2(m). Then the following are equivalent : (i) CT is nonnal, (ii) CT is unitary, is an isometry, (iii) C, (iv) C, is quasinormal, 42 11 Composition Operators on LP-Spaces (v) CT is a co-isometry. 7. If (X, 3'. rn) is a finite atomic measure space such that different atoms have different measures, then all five classes of composition operators on L2(rn) given above coincide with the singleton set containing the identity operator.

Hence there exists a sequence {C, fn} in the range of C, converging to P f in norm. Thus CTC; f = M ~ . T Pf for all f E L2(m). * Hence CTCT = M ~ . T P. (iii) If CT has dense range, then by (ii) P = I , the identity opcrator. TP . Conversely, suppose CTC; = M ~ . T P. , C T C ~ is an injection. Since C: and CT Ci have the same kernel, we have the desired result. This completes the proof of the theorem. fT o Note. Although the results in the above theorem look very simple, they have many useful applications, as will be seen in this section and several other later sections.

Hence by part (i), fT is zero almost everywhereon X I . Hence rnT-I(X1) = f T dm = 0 . Thus Xl x = T-yX2). 1 we get a contradiction. From the above theorems it is evident that the hope for existence of the compact composition operators lies in the case when the underlying measure space is atomic. The sequence spaces are examples of L2-spaces of atomic measure spaces. In the rest of this section we shall study compact composition operators on Hilbert spaces of sequences. Let T : N 4 N be a mapping and let n :n E E> 0.

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Composition Operators on Function Spaces by R.K. Singh and J.S. Manhas (Eds.)

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