Download Operator Theory, Systems Theory and Scattering Theory: by Daniel Alpay, Victor Vinnikov PDF

By Daniel Alpay, Victor Vinnikov

ISBN-10: 3764372125

ISBN-13: 9783764372125

Operator concept, approach thought, scattering thought, and the idea of analytic services of 1 advanced variable are deeply similar subject matters, and the relationships among those theories are good understood. whilst one leaves the surroundings of 1 operator and considers numerous operators, the location is way extra concerned. there isn't any longer a unmarried underlying idea, yet quite diversified theories, a few of them loosely hooked up and a few no longer hooked up in any respect. those numerous theories, which one can name "multidimensional operator theory", are themes of energetic and extensive examine. the current quantity includes a number of papers in multidimensional operator conception. themes thought of comprise the non-commutative case, functionality idea within the polydisk, hyponormal operators, hyperanalytic services, and holomorphic deformations of linear differential equations. the quantity can be of curiosity to a large viewers of natural and utilized mathematicians, electric engineers and theoretical physicists.

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Extra info for Operator Theory, Systems Theory and Scattering Theory: Multidimensional Generalizations

Sample text

41) we see that the operators (T1∗ , . . , Td∗ ) defined by ∗ Tj∗ Yv∗ e = Yvg e j ∗ extend to define a column contraction on L. By induction, T ∗u Yv∗ = Yvu ⊤ e or u⊤ Yv T = Yvu⊤ for all u ∈ Fd and e ∈ E, and ⎡ ⎤ L ⎢ ⎥ T1 . . Td : ⎣ ... ⎦ → L L is a contraction. Let (V1 , . . , Vd ) on L ⊃ L be the row-isometric dilation of (T1 , . . , Td ) (see [14]), so ⎡ ⎤ L ⎢ ⎥ V1 . . Vd : ⎣ ... ⎦ → L is isometric and PL V u = T u PL . Set L Xv,w = Yw PL V ∗v : L → E for v, w ∈ Fd . 29) for the last step.

Ud ⎣ ... ⎦ Yv∗1 Ud∗ Yvn ⎡ ⎤ Ygℓ v1 d ⎢ .. ⎥ = ⎣ . ⎦ Yv1 gℓ . . Yv∗n gℓ ℓ=1 ... ,d ≥ 0. 37) follows. Suppose now that W is a Cuntz weight. Then (UW,1 , . . , UW,d ) is row-unitary. It follows that (V1 , . . , Vd ) is row-unitary, and hence that (V1∗ , . . , Vd∗ )|L is a column isometry. 38) as asserted. 37) hold. Then we have Wv,w = Yw Yv∗⊤ where Yv : L → E. A. Ball and V. Vinnikov We may assume that L = closed span {Yv∗ E : v ∈ Fd }. 41) we see that the operators (T1∗ , . . , Td∗ ) defined by ∗ Tj∗ Yv∗ e = Yvg e j ∗ extend to define a column contraction on L.

29) we see that W [∗] is given by ⎧ ∗ ⎨ W(αv−1 )w⊤ ,β [∗] ∗ Wv,w;α,β = (Wα,β;v,w ) = ∗ ⎩ W ⊤ w ,β(vα−1 )⊤ if |α| ≥ |v|, if |α| ≤ |v|. 19) follows for W [∗] . 20) follows for W [∗] as well. 29) is [∗]-Haplitz as asserted. 29) motivates the introduction of the symbol W (z, ζ) for the [∗]-Haplitz operator W defined by Wv,w;∅,∅ ez v ζ w . W (z, ζ)e = (W e)(z, ζ) = v,w∈Fd For any [∗]-Haplitz W and given any α, β ∈ Fd , we have Wv,w;α,β ez v ζ w = W (ez α ζ β ) v,w ⊤ ⊤ = W ((S R )α (U R[∗] )β e) ⊤ ⊤ = (U R )α W ((U R[∗] )β e) ⊤ ⊤ = (U R )α (S R[∗] )β (W e).

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