Download Structure of Hilbert Space Operators by Chunlan Jiang PDF

By Chunlan Jiang

This ebook exposes the inner constitution of non-self-adjoint operators performing on advanced separable countless dimensional Hilbert area, via reading and learning the commutant of operators. a distinct presentation of the concept of Cowen-Douglas operators is given. The authors take the strongly irreducible operator as a simple version, and locate entire similarity invariants of Cowen-Douglas operators through the use of K-theory, complicated geometry and operator algebra instruments.

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Extra resources for Structure of Hilbert Space Operators

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I Proof For the first part of the proposition, if N = Yl (BNi,l

10, there exists a compact operator Kk, \\Kk\\ < fz, such that Mfc *" (J-k Tk + Kk cn C" cn> fc = l , 2 , - . 0 Since {/Ujtj^iCcftl, we can choose pairwise distinct numbers {A (fc) . j fc>l, j > l } in ft such that \fXk — A^KJ| < -^ for all j and fc. Thus there 36 Structure of Hilbert Space Operators is a compact operator Kk with \\Kk\\ < pr such that iW C" c \(2) Ak = Tk + Kk + Kk* i< 3 > C"i ^ = 1>2, ••• Furthermore, there is a compact operator C2,11C21 i < § satisfying • 'J4I Mi M2 M3' T + Cx + Co = 0 where C 2 = ^ ©(-Kk + ^fc)- Set if = C\ + C 2 , then K is compact and fc=i i ( * >J : J>1,/s>l} as {^k}kLi- ^ i s n o t ; difficult to see H^ll < e.

Proof Define X : H —> ran(T - A0) by Xx = (T - A0)x for x€H. Since ran(T — Ao) is closed and T — Ao is injective, X is invertible. , T\ran(T-\0)~T. 3) with respect to the ONB {efc}£L0. 7/||A|| < | , then (S + A)*e#i(fi), where S is the unilateral shift. Proof Since ||A|| < \ and |A0| < \, ind(S + A - A) = - 1 and min-ind(S + A - A) = 0 for |A| < - . Thus (S + A - A)* is surjective and nul(S + A - A)* = 1. In particular, ker(S + A- A0)* = {ae0 : aGC}. Therefore ran(S + A - A0) = \J{ek : fc>l} := Hx.

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