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Register a new userSpectral sets and distinguished varieties in the symmetrized bidisc
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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.
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We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaevs classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain [{(z_1,z_2)in mathbb{C} ^2 : 1+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+overline{z_2}))>0}] in Isaevs list. Isaev calls it $mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of $mathcal D_1$. Among the results on $mathcal D_1$, of particular interest is the fact that $mathcal D_1$ is a symmetrization. When we symmetrize (appropriately defined in the context in the last section) either $Omega_1$ or $mathcal{D}^{(2)}_1$ (Isaevs notation), we get $mathcal D_1$. These two domains $Omega_1$ and $mathcal{D}^{(2)}_1$ are in Isaevs list and he mentioned that these are biholomorphic to $mathbb{D} times mathbb{D}$. We produce explicit biholomorphisms between these domains and $mathbb{D} times mathbb{D}$.
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.
It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets.
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 $mathbb{B}_n$.
We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)otimes_{rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{sharp} = {(s,t) : stin E}$ and show that if $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{sharp}$ is a set of spectral synthesis for $A(G)otimes_{rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)$-bimodule maps with the dual of $A(G)otimes_{rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.
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