Theory

By Gelu Popescu

ISBN-10: 0821847104

ISBN-13: 9780821847107

During this quantity the writer reports noncommutative domain names $\mathcal{D}_f\subset B(\mathcal{H})^n$ generated by means of confident ordinary loose holomorphic features $f$ on $B(\mathcal{H})^n$, the place $B(\mathcal{H})$ is the algebra of all bounded linear operators on a Hilbert area $\mathcal{H}$. desk of Contents: advent; Operator algebras linked to noncommutative domain names; loose holomorphic services on noncommutative domain names; version concept and unitary invariants on noncommutative domain names; Commutant lifting and purposes; Bibliography. (MEMO/205/964)

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Extra info for Operator theory on noncommutative domains

Example text

Now, we can prove the following. 10. Let ϕ(W1 , . . , Wn ) = β∈F+ n Hardy algebra Fn∞ (Df ) and such that ϕ(W1 , . . , Wn ) ≤ 1. If (T1 , . . , Tn ) ∈ Df,r (H), where 0 ≤ r < 1, then |cβ |Tβ ≤ |c0 | + (1 − |c0 |2 ) β∈F+ n r . 1−r Proof. 1, since ϕ(W1 , . . , Wn ) ∈ Fn∞ (Df ), ∞ |β| |cβ |r |β| Wβ we have k=0 |β|=k |cβ |r Wβ < ∞. Therefore, the series β∈F+ n converges in the operator norm for 0 ≤ r < 1 and it is in the algebra An (Df ). Since 1 1 r T1 , . . 2, part(ii), implies |cβ |r |β| Wβ .

2, part(ii), implies |cβ |r |β| Wβ . 1, the operators {Wβ }|β|=k have orthogonal ranges and Wβ = √1 , β ∈ F+ n . Consequently, bβ |β|=k bβ Wβ Wβ∗ ≤ 1 for any Operator Theory on Noncommutative Domains k = 0, 1, . .. Hence, and using the fact that β∈F+ n 55 |cβ |2 b1β < ∞, which is due to the fact that ϕ(W1 , . .

Ii) The only normal elements in Fn∞ (Df ) are the scalars. (iii) Every element A ∈ Fn∞ (Df ) has its spectrum σ(A) = {0} and it is injective. (iv) The algebra Fn∞ (Df ) contains no non-trivial idempotents and no non-zero quasinilpotent elements. (v) The algebra Fn∞ (Df ) is semisimple. 24 Gelu Popescu (vi) If A ∈ Fn∞ (Df ), n ≥ 2, then σ(A) = σe (A). Proof. 9 part (ii), using the fact cα Wα be in that Fn∞ (Df ) = Fn∞ (Df ). To prove (ii), let ϕ(W1 , . . , Wn ) := α∈F+ n Fn∞ (Df ). 7), we have ϕ(W1 , .