Recent Advances in Operator Theory and Applications by Tsuyoshi Ando, Il Bong Jung, Woo Young Lee

By Tsuyoshi Ando, Il Bong Jung, Woo Young Lee

This quantity includes the lawsuits of the foreign Workshop on Operator conception and purposes (IWOTA 2006) held at Seoul nationwide collage, Seoul, Korea, from July 31 to August three, 2006. The distinct curiosity parts of this workshop have been Hilbert/Krein house operator thought, advanced functionality thought regarding Hilbert house operators, and platforms idea regarding Hilbert area operators. This quantity includes 16 study papers which mirror fresh advancements in operator thought and functions.

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Extra info for Recent Advances in Operator Theory and Applications

Sample text

D. 3)) deﬁnes a bounded operator from HU2 (Fd ) to HY2 (Fd ). The noncommutative Schur class Snc,d (U, Y) is deﬁned to consist of such multipliers S for which MS has operator norm at most 1: Snc,d (U, Y) = {S ∈ L(U, Y) z : MS : HU2 (Fd ) → HY2 (Fd ) with MS op ≤ 1}. 1 for this setting. We refer to [39, 40] for details. 3. Let S(z) ∈ L(U, Y) z be a formal power series in z = (z1 , . . , zd ) with coeﬃcients in L(U, Y). , MS : U z → Y z given by MS : p(z) → S(z)p(z) extends to deﬁne a contraction operator from HU2 (Fd ) into HY2 (Fd ).

A Hilbert space which we will denote by H equipped with an L(H)-valued ∗-representation π : A → L(H) of A. 4), viewed as a (L(E), C)-correspondence. Let us suppose also that E is a reproducing kernel correspondence. , H is a reproducing kernel Hilbert space of vector-valued functions on Ω × A, but with the additional wrinkle that there is also a representation a → π(a) of A on H with π(a)(f ⊗ e) = (a · f ) ⊗ e such that (π(a)(f ⊗ e))(ω , a ) = f (ω , a a) ⊗ e with reproducing kernel (in the sense of a vector-valued reproducing kernel Hilbert space) K(·, ·) of the special form K((ω , a ), (ω, a)) = K(ω , ω)[a∗ a ] for a completely positive kernel K : Ω × Ω → L(A, L(E)): for f ∈ H(K), e ∈ E and (ω, a) ∈ Ω × A, f , K(·, ω)[a]e H = f (ω, a), e E where K is completely positive.

3)) unless B is commutative. If both A and B have units, we also demand that the scalar multiplication on E is compatible with both the identiﬁcation λ → λ1A of C as a subalgebra of A and the identiﬁcation λ → λ1B of C as a subalgebra of B. This is consistent with demanding the additional axioms (λa) · e = a · (λe), (λe) · b = e · (λb) for the general case. The classical case is the one where E is a Hilbert space E, B = C and A = L(E) with the operations given by a · e = ae (the operator a acting on the vector e) e · b = be (scalar multiplication in E), e, f (the E Hilbert-space inner product).