Let Ω be an irreducible bounded symmetric domain of rank r in C d . Let K be the maximal compact subgroup of the identity component G of the biholomorphic automorphism group of the domain Ω. The group K consisting of linear transformations acts naturally on any d-tuple T = (T 1 , . . . , T d ) of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that T is K-homogeneous. In this paper, we obtain a model for a certain class of K-homogeneous d-tuple T as the operators of multiplication by the coordinate functions z 1 , . . . , z d on a reproducing kernel Hilbert space of holomorphic functions defined on Ω. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class, (iii) unitary equivalence and similarity of these d-tuples. In particular, we show that the adjoint of the d-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class B 1 (Ω). For an irreducible bounded symmetric domain Ω of rank 2, an explicit description of the operator d i=1 T * i T i is given. In general, based on this formula, we make a conjecture giving the form of this operator.
We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space H 2 (β) of formal power series in the variables z 1 , . . . , z m is spherically balanced if and only if there exist a Reinhardt measure μ supported on the unit sphere ∂B and a Hilbert space H 2 (γ ) of formal power series in the variable t such thatwhere f z (t) = f (tz) is a formal power series in the variable t. In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple M z on a spherically balanced space H 2 (β). We also observe that all spherical contractive multishifts on spherically balanced spaces admit the classical von Neumann's inequality.In the second half, we introduce and study a class of Hilbert spaces, to be referred to as G-balanced Hilbert spaces, where G = U(r 1 ) × U(r 2 ) × · · · × U(r k ) is a subgroup of U(m) with r 1 + · · · + r k = m. In the case in which G = U(m), G-balanced spaces are precisely spherically balanced Hilbert spaces.
Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In particular, we show that if A and B are commuting contractive d-tuples of operators such that B satisfies the matrix-version of von Neumann's inequality and (1, . . . , 1) is in the algebraic spectrum of B, then the tensor product A ⊗ B satisfies the von Neumann's inequality if and only if A satisfies the von Neumann's inequality. We also exhibit several families of operator-valued multishifts for which the von Neumann's inequality always holds.2010 Mathematics Subject Classification. Primary 47B37, Secondary 47A13.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.