New PDF release: A band method approach to a positive expansion problem in a

By Frazho A.E., Kaashoek M.A.

Show description

Read or Download A band method approach to a positive expansion problem in a unitary dilation setting PDF

Best nonfiction_1 books

Get The quality audit handbook: principles, implementation, and PDF

Written by way of auditors, each one with a large number of real-world event, this isn't a publication that offers with theories and rules as they exist on library cabinets. New to this variation is the improved insurance on ethics, the correlation of auditing to company functionality, and the growth into parts now not lined by way of the qualified caliber Auditor physique of information.

Extra resources for A band method approach to a positive expansion problem in a unitary dilation setting

Example text

All C J~f+. 17 Let B be an operator on K:. Then B-f~+ E Afa if and only if B is a commuting expansion of A with respect to U. Proof. Assume B - f~+ E HI. Then B = (B - f2+) + f~+ E Afa + Af+ C AT+. Hence B commutes with U and BK:+ C K:+. The fact that (B - f~+)K:+ C K:+ E3 7/ implies that Pn(B - f~+)lT/= 0. 10. So P n B [ 7 / = A, and thus B is a commuting expansion of A with respect to U. To prove the converse, assume that B is a commuting expansion of A with respect to U. Then B EAf+ and P n B I T / = A.

12 Let {A; T, U} be the data for a commuting expansion problem, and assume that A + A* is strictly positive. 57) where G is an arbitrary operator in the set C(S+, S_, U). 54). 57) uniquely determine each other. The proof of this theorem uses the following lemmas. 13 Let F be an operator on ]~, and assume that F + F* is strictly positive. Then F is an invertible operator. Moreover, i f M is an invariant subspaee for F, then the operator F[ AA is an invertible operator on 3,l. In this case, F maps AJ one to one and onto At.

5) PnBI74 = X-" T k K E + * T *~ 9 k=O To see this let B()~) = ~ = o AkBk be the Taylor series expansion for/~. ~T*)-1 = v'r A k E +* T *k for A in the unit disc. Fix g E 7/. ~1-- 1) belongs to L~ and by matching coefficients we see that for n > 0 the coefficient of e ~n~ in the Fourier expansion of G is given by ~ = o B~E~-T*(k+~)" Using K* x-'~176 B 'kE *+ T *k Z-~k=0 it follows that PH=(E+)B()~) E+(I - s = E ;~K*T*~g = K * ( I - AT*)-lg. n-~O Recall that (~4g)(A) = E ~ ( I - AT*)-lg for IA[ < 1.

Download PDF sample

A band method approach to a positive expansion problem in a unitary dilation setting by Frazho A.E., Kaashoek M.A.


by James
4.0

Rated 4.54 of 5 – based on 4 votes

Related posts