On the Multiplication Tuples Related to Certain Reproducing Kernel Hilbert Spaces

Multivariable isometries related to certain convex domains

“…This negative result can actually be extended to the multiplication tuples M σp,z and M νp,z associated with the domains Σ p . The next proposition generalizes [6,Proposition 3.4 (d)] with an analogous proof; a complete proof is presented here for the reader's convenience. Proposition 4.12.…”

“…. , T n ) of commuting operators T i in B(H) is said to be essentially normal if the operators T * i T j − T j T * i are compact for all i and j, while T is said to be cyclic if there exists a vector f in H (referred to as a cyclic vector for T ) such that the linear span ∨{T k1 1 T k2 2 • • • T kn n f : k i are non-negative integers} is dense in H. There exist several interesting results in the literature on subnormal operator tuples (and in particular on essentially normal and/or cyclic subnormal operator tuples) having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in C n (refer, for example, to [4], [6], [15], [16], [17], [18], [20], [21], [48]). These results are largely manifestations of the functional calculus for subnormal operator tuples thriving upon some elegant function-theoretic results valid in the context of those two types of domains.…”

Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

“…Taking supremum over all unit vectors g, g ∈ E, the claim stands verified. Combining (7) with the assumption (5), we obtain for any w ∈ Ω,…”

“…A bounded open connected subset Ω of C d is said to be an admissible domain if it has the following property: For any bounded holomorphic function φ : Ω → C, there exists a sequence {p n } ∞ n=1 of polynomials such that • for some M > 0, p n ∞,Ω M φ ∞,Ω for every integer n 1, • p n (w) converges to φ(w) as n → ∞ for every w ∈ Ω. It is well-known that if a bounded domain Ω has polynomially convex closure in C d , then Ω is admissible provided it is star-shaped or strictly pseudoconvex with C 2 boundary (see, for instance, [37, Proof of Theorem 4], [7,Lemma 2.2]). In what follows, we also need the notion of vector-valued holomorphic function f : Ω → Z, where Ω is a domain in C d and Z is a normed linear space.…”

“…This provides several new classes of examples and recovers special cases of various known results in one and several variables, see [47,Theorem 3], [50, Section 10, Proposition 37], [10,Theorem 15], [12], [33, Theorem 5.2], [49,Theorem 4.2], [37, Section 0, Theorem 4], [4, Theorem A] (cf. [18,Proposition 4.4], [43,Theorem 3], [37,Theorem 3], [34,Theorem 3.1], [26, Corollary 3.7], [7,Theorem 1.2], [27,Theorem 2.11], [30,Corollary 7]). It is worth noting that the techniques employed in the proofs of Theorems 3.1 and 4.1 have some common features (e.g.…”