By de la Harpe P., Jones V.
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COMPACT AND HILBERT-SCHMIDT OPERATORS (ii) Let p denote the orthogonal projection of H onto E: Then pa : H ! E is a bounded operator which is onto. 5 that there exists a non empty ball in E which is contained in pa(H(1)): Hence dimC (E ) < 1 by (i). 12. Examples. Let H be in nite dimensional. The identity operator on H is not compact. ) Consequently, any invertible operator on H is not compact. 15, which constitutes an epsilon of spectral theory. 13. Observation. 14. Lemma. Let a be a compact self-adjoint operator on H: Then one at least of the numbers kak ;kak is an eigenvalue of a: Proof.
Spectrum of an element in a C -algebra T Let A be a C -algebra with unit. We denote by the unit circle of the complex plane. 10 4. 20. Proposition. (i) For each a 2 A the spectrum of a is (a): (ii) If a 2 A is self-adjoint, its spectrum is in : (iii) If u 2 A is unitary, its spectrum is in : Proof. (i) For 2 the element ; a is invertible (say with inverse b), if and only if ; a is invertible (with inverse b ). 14 implies that jx + i(y + t)j2 = x2 + (y + t)2 kak2 + t2 and this inequality can also be written as R 2yt kak2 ; x2 ; y2 : As this has to hold for all t 2 one has y = 0: (iii) Let 2 (u): Observe that 6= 0 because u is invertible, and that ;1 2 (u;1 ) because ;1 ; u;1 = ; ;1( ; u)u;1 is not invertible.
Let H be a separable in nite dimensional Hilbert space given together with an orthonormal basis ( j )j N and let ( j )j N be a sequence of complex numbers which converges to zero. Then the diagonal operator a de ned by 0 0 0 0 2 2 aj= j j for all j 2 N is compact, because it is a norm limit of operators of nite rank. 15 that any compact operator which is also self-adjoint is of this form (with real j 's). For each function f 2 L2( 0 1] 0 1]) the operator af with kernel f is de ned by (af ( )) (x) = Z1 0 f (x y) (y)dy for all 2 L2( 0 1]) and x 2 0 1]: It is a compact operator on L2( 0 1]): See Problem 135 in Hal].
An introduction to C-star algebras by de la Harpe P., Jones V.