Linear Operators and their Spectra (Google eBook)

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Cambridge University Press, Apr 26, 2007 - Mathematics
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This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, and contains many illustrative examples and exercises. Fredholm theory, Hilbert-Schmidt and trace class operators are discussed, as are one-parameter semigroups and perturbations of their generators. Two chapters are devoted to using these tools to analyze Markov semigroups. The text also provides a thorough account of the new theory of pseudospectra, and presents the recent analysis by the author and Barry Simon of the form of the pseudospectra at the boundary of the numerical range. This was a key ingredient in the determination of properties of the zeros of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non-self-adjoint Schrodinger operators are described. The text concludes with a description of the surprising spectral properties of the non-self-adjoint harmonic oscillator.
  

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Contents

1
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2
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3
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4
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5
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53 Orthogonal projections
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Prove also that the second inequality becomes an equality if
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Since t 0 is arbitrary this implies that Rez
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and a oneparameter unitary group Ut on such that
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11
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rMeafMA7
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From now on we often do not specify whether an
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Problem 6311 Formulate and prove a vectorvalued version of Theorem
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7
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Proof We start with the observation that
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define two oneparameter semigroup s on which commute with
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jLewdxa if dx a n
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Page 3 - This rxxik follows the convention that inner products are linear in the first variable and conjugate linear in the second variable. 2Some books use the words "completely continuous" in place of "compact
Page 2 - X, then there exists a continuous function f: X — >- [0, 1] such that f(x) - 0 for all x € A and f(x) - 1 for all x € B.
Page 14 - Every continuous linear operator A from the Banach space B^ to the Banach space #2 defines an adjoint operator A , which takes II -^ into fij, by the equation (A*t) (/) = l(Af) (fe Bv I e B2'). The characteristics of A" induced by those of A are well known.

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About the author (2007)

E. Brian Davies is a Professor of Mathematics at King's College London and a Fellow of the Royal Society. This is his seventh book.

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