Holomorphic functions on the symmetrized bidisk - realization, interpolation and extension

There are three new things in this talk about the open symmetrized bidisk $\mathbb G = \{ (z_1+z_2, z_1z_2) : |z_1|, |z_2| \u3c 1\}$. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extension Theorem: A characterization is obtained of those subsets $V$ of the open symmetrized bidisk $\mathbb G$ that have the property that every function $f$ holomorphic in a neighbourhood of $V$ and bounded on $V$ has an $H^\infty$-norm preserving extension to the whole of $\mathbb G$. \end{enumerate

Admissible fundamental operators

Let $F$ and $G$ be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigate when there is a $\Gamma$-contraction $(S,P)$ such that $F$ is the fundamental operator of $(S,P)$ and $G$ is the fundamental operator of $(S^*,P^*)$. Theorem 1 puts a necessary condition on $F$ and $G$ for them to be the fundamental operators of $(S,P)$ and $(S^*,P^*)$ respectively. Theorem 2 shows that this necessary condition is sufficient too provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for $\Gamma$-contractions are then applied to tetrablock contractions to figure out when two pairs $(F_1, F_2)$ and $(G_1, G_2)$ acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction $(A, B, P)$ and its adjoint $(A^*, B^*, P^*)$ respectively. This is the content of Theorem 4

On the geometry of the symmetrized bidisc

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 Isaev's 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 Isaev's 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 $\mathcal{D}^{(2)}_1$ (Isaev's notation), we get $\mathcal D_1$. These two domains $\Omega_1$ and $\mathcal{D}^{(2)}_1$ are in Isaev's list and he mentioned that these are biholomorphic to $\mathbb{D} \times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $\mathbb{D} \times \mathbb{D}$.Comment: 22 pages, Accepted in Indiana University Mathematics Journa

On certain Toeplitz operators and associated completely positive maps

We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to $L^\infty$ of a certain compact subset of $\mathbb C^n$. Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do not hold in general for $n>2$.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1706.0346

The Defect Sequence for Contractive Tuples

We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.Comment: 16 Pages. To appear in Linear Algebra and its Application

On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel

Given a domain $\Omega$ in $\mathbb{C}^m$, and a finite set of points $z_1,z_2,\ldots, z_n\in \Omega$ and $w_1,w_2,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the \textit{Pick interpolation problem} asks when there is a holomorphic function $f:\Omega \rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $\Omega=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to a domain $\Omega$ in $\mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.Comment: arXiv admin note: text overlap with arXiv:1812.00147, Accepted in Studia Mathematic

Kernels with complete Nevanlinna-Pick factors and the characteristic function

The Sz.-Nagy Foias charateristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna- Pick (CNP) kernels. Indeed, an irreducible unitarily invariant kernel $k$ on the $d$-dimensional Euclidean unit ball admits a characteristic function if and only if $k$ is a CNP kernel. We are intrigued by recent constructions of the characteristic function for kernels which are not CNP. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector valued Drury-Arveson space (thus the Hardy space when $d = 1$). We show that the construction of a characteristic function is always possible when $k$ has a CNP factor $s$.Comment: 16 Page

Completely bounded kernels

We introduce completely bounded kernels taking values in L(A,B) where A and B are C*-algebras. We show that if B is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and that this is automatic when the index set is countable.Comment: 22 pages. Fixed oversight in previous version. To appear in Acta Scientiarum Mathematicarum (Szeged) for the 100th anniversary of the birth of Bela Sz.-Nag

Coincidence of Schur multipliers of the Drury-Arveson space

In a purely multi-variable setting (i.e., the issues discussed in this note are not interesting in the single variable operator theory setting), we show that the coincidence of two operator valued Schur class multipliers of a certain kind on the Drury-Arveson space is characterized by the fact that the associated colligations (or a variant, obtained canonically) are `unitarily coincident' in a sense to be made precise in the last section of this article