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A triple of commuting operators for which the closed tetrablock E ‾ is a spectral set is called a tetrablock contraction or an E -contraction. The set E is defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . We show that every E -contraction can be uniquely written as a direct sum of an E -unitary and a completely non-unitary E -contraction. It is analogous to the canonical decomposition of a contraction operator into a unitary and a completely non-unitary contraction. We produce a concrete operator model for such a triple satisfying some conditions. [ABSTRACT FROM AUTHOR]

We show by a counter example that rational dilation fails on the tetrablock, a polynomially convex and non-convex domain in C 3 defined as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 0 whenever | z | ≤ 1 , | w | ≤ 1 } . A commuting triple of operators ( T 1 , T 2 , T 3 ) for which the closed tetrablock E ‾ is a spectral set, is called an E -contraction. For an E -contraction ( T 1 , T 2 , T 3 ) , the two operator equations T 1 − T 2 ⁎ T 3 = D T 3 X 1 D T 3 and T 2 − T 1 ⁎ T 3 = D T 3 X 2 D T 3 , D T 3 = ( I − T 3 ⁎ T 3 ) 1 2 , have unique solutions A 1 , A 2 on D T 3 = Ran ‾ D T 3 and they are called the fundamental operators of ( T 1 , T 2 , T 3 ) . For a particular class of E -contractions, we prove it necessary for the existence of rational dilation that the corresponding fundamental operators A 1 , A 2 satisfy (0.1) A 1 A 2 = A 2 A 1 and A 1 ⁎ A 1 − A 1 A 1 ⁎ = A 2 ⁎ A 2 − A 2 A 2 ⁎ . Then we construct an E -contraction from that particular class which fails to satisfy (0.1) . We produce a concrete functional model for pure E -isometries, a class of E -contractions analogous to the pure isometries in one variable. The fundamental operators play the main role in this model. [ABSTRACT FROM AUTHOR]

Abstract: We show that for every pair of matrices , having the closed symmetrized bidisc Γ as a spectral set, there is a one dimensional complex algebraic variety Λ in Γ such that for every matrix valued polynomial , The variety Λ is shown to have the determinantal representation where F is the unique matrix of numerical radius not greater than 1 that satisfies When is a strict Γ-contraction, then Λ 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. [Copyright &y& Elsevier]

Abstract: For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation where X is a bounded operator on with numerical radius of X being not greater than 1. A pair of bounded operators which has the domain as a spectral set, is called a Γ-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction . This allows us to construct an explicit Γ-isometric dilation of a Γ-contraction . We prove the other way too, i.e., for a commuting pair with and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that is a Γ-contraction. We show that for a pure Γ-contraction , there is a bounded operator C with numerical radius not greater than 1, such that . Any Γ-isometry can be written in this form where P now is an isometry commuting with C and . Any Γ-unitary is of this form as well with P and C being commuting unitaries. Examples of Γ-contractions on reproducing kernel Hilbert spaces and their Γ-isometric dilations are discussed. [Copyright &y& Elsevier]

An n -tuple (S 1 , ... , S n − 1 , P) of commuting Hilbert space operators, for which the closed symmetrized polydisc Γ n = { (∑ i = 1 n z i , ∑ 1 ≤ i < j ≤ n z i z j , ... , ∏ i = 1 n z i) : | z i | ≤ 1 , i = 1 , ... , n } is a spectral set, is called a Γ n -contraction. Costara characterized a point in Γ n (or in its interior G n) via a rational function f ˜ s _ in the following way: s _ = (s 1 , ... , s n − 1 , p) ∈ Γ n (or ∈ G n) ⇔ | f ˜ s _ (z) | ≤ 1 (or < 1) , ∀ z ∈ D ‾ , where f ˜ s _ is the continuous extension of Costara's function f s _ defined on D by f s _ (z) = n (− 1) n p z n − 1 + (n − 1) (− 1) n − 1 s n − 1 z n − 2 + ... + (− s 1) n − (n − 1) s 1 z + ... + (− 1) n − 1 s n − 1 z n − 1 . In this paper, we investigate an operator theoretic analogue of this result. We prove that for a Γ n -contraction only the forward implication holds and the converse holds for subnormal Γ n -contractions. We provide a counterexample to show that the converse does not hold if we omit the subnormality condition. We obtain a few new characterizations of Γ n -isometries and Γ n -unitaries, which are analogues for Γ n -contractions of the isometries and unitaries respectively in one variable operator theory. We show that (s 1 , ... , s n − 1 , p) ∈ Γ n if and only if (s 1 , ... , s n − 1 , p , 0 , ... , 0 ︸ k − t i m e s) ∈ Γ n + k , but, the operator theoretic analogue of this result does not hold in general. We prove that if (S 1 , ... , S n − 1 , P) is a Γ n -contraction then (S 1 , ... , S n − 1 , P , 0 , ... , 0 ︸ k − t i m e s) is a Γ n + k -contraction and construct a counterexample to show that the converse is not always true. [ABSTRACT FROM AUTHOR]