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

By Daniel Alpay, Victor Vinnikov

Operator conception, approach thought, scattering concept, and the idea of analytic features of 1 complicated variable are deeply comparable themes, and the relationships among those theories are good understood. while one leaves the environment of 1 operator and considers numerous operators, the placement is far extra concerned. there isn't any longer a unmarried underlying conception, yet fairly varied theories, a few of them loosely attached and a few now not attached in any respect. those a number of theories, which you can actually name "multidimensional operator theory", are themes of energetic and in depth examine. the current quantity includes a choice of papers in multidimensional operator concept. themes thought of comprise the non-commutative case, functionality concept within the polydisk, hyponormal operators, hyperanalytic services, and holomorphic deformations of linear differential equations. the amount should be of curiosity to a large viewers of natural and utilized mathematicians, electric engineers and theoretical physicists.

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

Example text

A. Ball and V. Vinnikov We introduce the class of free atomic representations of Od studied by Davidson and Pitts [9] in the framework used by Bratteli and Jorgensen [7] for the study of the special case of permutative representations. We let K be a Hilbert space with orthonormal basis {ei : i ∈ I} indexed by some index set I. Let σ = (σ1 , . . , σd ) be a function system of order d on I; by this we mean that each σk : I → I is an injective function from I to I σk (i) = σk (i′ ) for some i, i′ ∈ I =⇒ i = i′ for each k = 1, .

An application of this general principle tells us that a subset I ′′ ⊂ I has the property that the associated subspace H := closed span {ei : i ∈ I ′′ } ⊂ K is reducing for U σ,λ if and only if I ′′ has the property d I ′′ = σk (I ′′ ). 10) is I ′′ = ∅ and I ′′ = I. Note that this latter statement is independent of the choice of i ∈ I; thus ei is ∗-cyclic for U σ,λ for some i ∈ I if and only if ei is ∗-cyclic for each i ∈ I. In general I partitions into ergodic subsets: I = ∪{Iα : α ∈ A} where Iα ∩ Iα′ = ∅ for α = α′ , Iα is invariant under both σ and σ −1 , and the restriction of σ to Iα is ergodic for each α ∈ A.

D} such that i = σk (i′ ) for some (necessarily unique) i′ ∈ I. Then we write i′ = σk−1 (i). For v ∈ Fd a word of the form v = gkn · · · gk1 (with k1 , . . , kd ∈ {1, . . , d}), we define σ v as the composition of maps σ v = σkn ◦ · · · ◦ σk1 . If k ′ is an element of {1, . . , d} not equal to this particular k, then we say that σk−1 (i) is undefined (or empty). More generally, given i ∈ I and a natural number n, there is a unique word v = gkn . . gk1 in Fd of length n so that σ v (i′ ) = i for some (necessarily unique) i′ ∈ I.

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