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A Cauchy-Type Problem for the Diffusion-Wave Equation with by Voroshilov A. A., Kilbas A. A.

By Voroshilov A. A., Kilbas A. A.

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Extra resources for A Cauchy-Type Problem for the Diffusion-Wave Equation with Riemann-Liouville Partial Derivative

Example text

Zd are noncommuting indeterminates rather than commuting variables. 3: The proof of (3) =⇒ (2) =⇒ (1 ) =⇒ (1 ) =⇒ (1) formally goes through in the same was as the classical case. Let us just note that (1 ) =⇒ (1 ) involves viewing MS : HU2 (Fd ) → HY2 (kd ) as MS = S(S) where S = (S1 , . . , Sd ) are the left creation operators of multiplication by zj on the left on the Fock space H 2 (Fd ). From the assumption (1 ), we know that S(rS) ≤ 1 for each r < 1 and hence MS = limr↑1 S(rS) ≤ 1 as well.

N) where the block entry Tξ (n) Tξ : E ⊗n → E ⊗n+1 is given by : ξn ⊗ · · · ⊗ ξ1 → ξ ⊗ ξn ⊗ · · · ⊗ ξ1 . The operator Tξ is also in La (F 2 (E)). In summary, both Tξ and ϕ∞ (a) are Amodule maps with respect to the right A-action on F 2 (E) for each ξ ∈ E and a ∈ A. Moreover, one easily checks that ϕ∞ (a)Tξ = Taξ = Tϕ(a)ξ and Tξ ϕ∞ (a) = Tξa for each a ∈ A and ξ ∈ E. ∞ We let F (E) denote the weak-∗ closed algebra generated by the collection of operators {ϕ∞ (a), Tξ : a ∈ A and ξ ∈ E} in the W ∗ -algebra La (F (E)) – we prefer this notation over the notation H ∞ (E) used for this object in [31, 33].

4) Multivariable Generalizations of the Schur Class 31 with right C-action given by (e ⊗ f ) · c = e ⊗ (f · c), and with C-valued inner product ·, · e ⊗ f, e ⊗ f E⊗F = E⊗F given by e, e E · f, f F . It is a straightforward exercise to verify that the balanced tensor-product construction is well defined. For example the computation (e · b) ⊗ f, (e · b ) ⊗ f = b ∗ · e, e · b · f, f = e, e · b · f, b · f = e ⊗ (b · f ), e ⊗ (b · f ) shows that the E ⊗ F -inner product is well defined. 2. Bounded linear operators between direct sum correspondences admit operator matrix decompositions in precisely the same way as in the Hilbert space case (B = C), while adjointability of such an operator corresponds to the operators in the decomposition being adjointable.

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