By de la Harpe P., Jones V.
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5. De nition. A bounded operator a : H ! H is compact if the image by a of the 0 closed unit ball H(1) in H is relatively compact in H : The linear space of all compact operators from H to H is denoted by K(H H ): We write K(H) instead of K(H H): 0 0 0 Typeset by AMS-TEX 1 2 3. 6. Remark. ) Let a : H ! H be a bounded operator. Then 0 a(H(1)) is relatively compact in H () a(H(1)) 0 is compact. Proof. ) We introduce the weak topology on H of which a basis of neighbourhoods of 0 consists of the nite intersections of the sets of the form V = f 2 H j jh j ij < 1g where 2 H 6= 0: It follows from Tychono 's theorem that the unit ball H(1) is compact with respect to this weak topology.
18. 22. Corollary. 23. Remark. There are elements a 2 A such that (a) < kak : The simplest example is probably the nilpotent operator a = 00 10 2 B( 2 ): Another example is the Volterra C integration operator V de ned on L2 ( 0 1]) by (V f )(x) = Z 0 x f (t)dt: It is a quasi-nilpotent operator, namely one with spectral radius equal to zero. See Problem 80 and Solution 147 in Hal]. 24. Theorem (Continuous functional calculus for bounded self-adjoint operators). 8). Then there exists a unique morphism of C -algebra ( C ( (a)) ;!
Exercice. In case dimC (H) < 1 observe that hajbi = trace(a b) this carries over to the in nite dimensional case in terms of the canonical trace, de ned on the appropriate two-sided ideal of trace class operators. 21. Remark. For the theory of operator ideals in general and of Hilbert-Schmidt operators in particular, the classical book is that by R. Schatten Sch]. For one (out of many) more recent exposition of Hilbert-Schmidt operators, see Sim]. 22. Example. 10 by Z1 (af ( )) (x) = 0 f (x y) (y)dy is a Hilbert-Schmidt operator see Problems 135-6 in Hal].