On the Joint Spectrum and H ∞ -Functional Calculus for Pairs of Commuting Contractions

On the Existence of Invariant Subspaces and Reflexivity of N-Tuples of Operators

“…It can be seen that it is minimal in the standard meaning; for the details, see [53]. We say that a pair {T 1 , T 2 } ⊂ L(H) is diagonally extendable if there exists a Hilbert space K ⊃ H and a minimal joint coisometric extension {B 1 , B 2 } ⊂ L(H) of {T 1 , T 2 } such that, for either j = 1 or j = 2, if K is decomposed as K = S j ⊕R j , relative to which the matrix for B j has the form…”

“…us recall the result of[53, Theorem 2.5] and[68, Lemma 1] as R 2 has no part of uniform multiplicity ℵ 0 , (c) T 1 and T 2 doubly commute. Let {T 1 , T 2 } ⊂ L(H) be a pair of commuting contractions.…”

Expanding the joint spectrum of pairs of commuting contractions

“…The following lemma is a known result (cf., [9], p.24, [18]): Proof. We may assume without loss of generality that E = S, where S is the semicircle of T centered at 1.…”

“…0 -pair that has a large joint spectrum is presented in [18] (i.e., it is shown that T 2 ⊆ σ H (S 1 , S 2 )). The construction is based upon a conformal mapping of a completely nonunitary contraction T with an already large spectrum (σ(T ) = D).…”