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Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of $n$-fold tensor products of Drury-Arveson spaces.
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We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $Lambda$ in $Gamma$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Lambda$. The variety $Lambda$ is shown to have a particular determinantal representation, related to the so-called fundamental operator of the pair (S,P). When (S,P) is a strict $Gamma$-contraction, then $Lambda$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.
In the spirit of Grothendiecks famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Frechet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optim
In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This is a significant improvement of the results of the first author and K. Wang, where the same result is shown to hold for polynomial-generated principal submodules of the Bergman module on the unit ball $mathbb{B}_n$ of $mathbb{C}^n$. As a consequence of our main result, we prove that the submodule of $L_a^2(mathbb{B}_n)$ consisting of functions vanishing on a pure analytic subsets of codimension $1$ is $p$-essentially normal for all $p>n$.
In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $partial{mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice properties and intersecting nicely with $partialmathbb{B}_n$, then the Geometric Arveson-Douglas Conjecture holds on this variety. We obtain this result by applying a result by Suarez, which allows us to localize the problem. Our result then follows from the simple case when the two varieties are intersection of linear subspaces with $mathbb{B}_n$.
A closed subspace $mathcal{M}$ of the Hardy space $H^2(mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $mathcal{M}$ containing $z_1 - varphi(z_2)$ is Hilbert-Schmidt, where $varphi$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.
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