Linear Operators and their SpectraThis 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. |
Contents
Section 1 | 35 |
Section 2 | 67 |
Section 3 | 74 |
Section 4 | 99 |
Section 5 | 133 |
Section 6 | 135 |
Section 7 | 140 |
Section 8 | 153 |
Section 19 | 296 |
Section 20 | 300 |
Section 21 | 301 |
Section 22 | 307 |
Section 23 | 308 |
Section 24 | 310 |
Section 25 | 313 |
Section 26 | 325 |
Section 9 | 163 |
Section 10 | 182 |
Section 11 | 190 |
Section 12 | 197 |
Section 13 | 209 |
Section 14 | 210 |
Section 15 | 220 |
Section 16 | 226 |
Section 17 | 245 |
Section 18 | 285 |
Section 27 | 355 |
Section 28 | 362 |
Section 29 | 375 |
Section 30 | 377 |
Section 31 | 380 |
Section 32 | 399 |
Section 33 | 408 |
Section 34 | 420 |
Section 35 | 427 |
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Common terms and phrases
acting in L2 algebra analytic function applying assume Banach space bounded linear operator bounded operator closed operator compact operator complete constant continuous function Corollary countable deduce denote differential operator domain eigenvalue equivalent essential spectrum EssSpec Example exists following theorem formula Fourier transform Fredholm operator functions f given Hence Hilbert space implies invariant invertible irreducible isometric L2 RN Lemma linear subspace Lp RN Lp spaces Lp X dx Markov operator Markov semigroup matrix measure n→lim non-negative norm continuous norm convergent numerical range one-one one-parameter contraction semigroup one-parameter group one-parameter semigroup operator acting operator on L2 orthonormal perturbation polynomial positive Problem Proof Prove pseudospectra satisfies Schrödinger operators self-adjoint operator semigroup acting sequence Spec spectral theorem subset Suppose topology Ttis unbounded unitary vector
Popular passages
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.