Appendix 1 Operators on Hilbert Space
Appendix 1 Operators on Hilbert Space We collect in this appendix necessary information on linear operators on Hilbert space. We give here almost no proofs and we give references for more detailed information. 1. Singular Values and Operator Ideals Let T be a bounded linear operator from a Hilbert space 11. to a Hilbert space K. The singular values Sn (T), n E Z+, of T are defined by sn(T) ~f inf{IIT - KII: K: 11. --t K, rankK::; n}, where rankK stands for the rank of K. Clearly, the sequence {sn(T)}n?:o is nonincreasing and its limit soo(T) ~f lim sn(T) n-+oo is equal to the essential norm of T, which is by definition IITlle = inf{IIT - KII: K E C(li, K)}, where C(1i, K) is the space of compact operators from 11. to K. It is easy to see that if Tl and T2 are operators from 11. to K, then sm+n(T1 + T 2 ) ::; sn(Td + sm(T2 ), m, nE Z+. If A, T, and B are bounded linear Hilbert space operators such that the product ATB makes sense, then it can easily be seen that 706 Appendix 1. Operators on Hilbert Space If T is a self-adjoint operator and D is a nonnegative rank one operator, and T # = T + D, then sn(T#) ~ sn(T) ~ sn+l(T#) ~ sn+l(T) for any nE 1:+. The operator T is compact if and only if lim sn(T) = O. If T is a n-HXl compact operator from 1l to K, it admits a Schmidt expansion Tx = L sn(T)(x, fn)gn, x E 1l, n::::O where {fn}n::::o is an orthonormal sequence in 1l and {gn}n::::O is an or thonormal sequence in K.
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