Abstract. There are three new things in this paper about the open symmetrized bidisk G = {(z 1 + z 2 , z 1 z 2 ) : |z 1 |, |z 2 | < 1}. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved.(1) The Realization Theorem: A realization formula is demonstrated for every f in the norm unit ball of H ∞ (G). (2) 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. (3) The Extension Theorem: A characterization is obtained of those subsets V of the open symmetrized bidisk G that have the property that every function f holomorphic in a neighbourhood of V and bounded on V has an H ∞ -norm preserving extension to the whole of G.
This is called the tetrablock. This paper constructs explicit boundary normal dilation for a triple (A, B, P) of commuting bounded operators which has E as a spectral set. We show that the dilation is minimal and unique under a certain natural condition. As is well-known, uniqueness of minimal dilation usually does not hold good in several variables, e.g., Ando's dilation is known to be not unique, see Li and Timotin (J Funct Anal 154:1-16, 1998). However, in the case of the tetrablock, the third component of the dilation can be chosen in such a way as to ensure uniqueness.
The symmetrized bidisc has grabbed a great deal of attention of late because of its rich structure both in the context of function theory and in the context of operator theory. Toeplitz operators on this domain have not been discussed so far. The distinguished boundary bΓ of the symmetrized bidisc is topologically identifiable with the Mobius strip and it is natural to consider bounded measurable functions there. In this article, we show that there is a natural Hilbert space H 2 (G). We describe three isomorphic copies of this space. The L ∞ functions on bΓ induce Toeplitz operators on this space. Such Toeplitz operators can be characterized through a couple of relations that they have to satisfy with respect to the co-ordinate multiplications on the space H 2 (G) which we call the Brown-Halmos relations. A number of results are obtained about the Toeplitz operators which bring out the similarities and the differences with the theory of Toeplitz operators on the disc as well as the bidisc. We show that the Coburn alternative fails, for example. However, the compact perturbations of Toeplitz operators are precisely the asymptotic Toeplitz operators. This requires us to find a characterization of compact operators on the Hardy space H 2 (G). The only compact Toeplitz operator turns out to be the zero operator.Although operator theory on the symmetrized bidisc has now been studied for quite some time, often there are occasions when one has to develop a result that one needs. Such is the case we encountered in the study of dual Toeplitz operators in the last section of this paper. In that section, we produce a new result about a family of commuting Γisometries. Just like a Toeplitz operator is characterized by the Brown-Halmos relations with respect to the co-ordinate multiplications, an arbitrary bounded operator X which satisfies the Brown-Halmos relations with respect to a commuting family of Γ-isometries is a compression of a norm preserving Y acting on the space of minimal Γ-unitary extension of the family of isometries. Moreover, if X commutes with the Γ-isometries, then Y is an extension and commutes with the minimal Γ-unitary extensions. Thus, it is a commutant lifting theorem. This result is then applied to characterize a dual Toeplitz operator.2010 Mathematics Subject Classification. 47A13, 47A20, 47B35, 47B38, 46E20, 30H10.
A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator T can be lifted to an isometry V . A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced by a certain domain contained in C 3 (called the tetrablock), (ii) the contraction operator T is replaced by a commutative triple (T 1 , T 2 , T ) of Hilbert-space operators having E as a spectral set (a tetrablock contraction) . The rational dilation question for this setting is whether a tetrablock contraction (T 1 , T 2 , T ) can be lifted to a tetrablock isometry (V 1 , V 2 , V ) (a commutative operator tuple which extends to a tetrablock-unitary tuple (U 1 , U 2 , U )-a commutative tuple of normal operators with joint spectrum contained in the distinguished boundary of the tetrablock). We discuss necessary conditions for a tetrablock contraction to have a tetrablock-isometric lift. We present an example of a tetrablock contraction which does have a tetrablock-isometric lift but violates a condition previously thought to be necessary for the existence of such a lift. Thus the question of whether a tetrablock contraction always has a tetrablock-isometric lift appears to be unresolved at this time. 2010 Mathematics Subject Classification. Primary: 47A13. Secondary: 47A20, 47A25, 47A56, 47A68, 30H10. K = span{r(U)h : h ∈ H, r ∈ Rat(Ω)} and then set V = U| K . For any r ∈ Rat(Ω) and h ∈ H, we have P K⊖H r(U)h = r(U)h − r(T)h = r(V)h − r(T)h and hence we see that K ⊖ H can be taken to have the form K ⊖ H = span{r(V)h − r(T)h : h ∈ H, r ∈ Rat(Ω)}. Then the computation, for r, q ∈ Rat(Ω) and h, h ′ ∈ H, q(V)(r(V) − r(T))h, h ′ K = q(U)(r(U) − r(T))h, h ′ K
This note constructs an explicit normal boundary dilation for a commuting pair (S, P ) of bounded operators with the symmetrized bidiskas a spectral set. Such explicit dilations had hitherto been constructed only in the unit disk [11], the unit bidisk [3] and in the tetrablock [6]. The dilation is minimal and unique under a suitable condition. This paper also contains a natural example of a Γ-isometry. We compute its associated fundamental operator.Date: April 4, 2018. MSC2010: Primary:47A20, 47A25.
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