A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. Norms ‖T n‖ converge to zero arbitrarily fast. Let H be a complex separable Hilbert space and let B(H) denote the algebra of all continuous linear operator on H. If T ∈ B(H) then {T}′ = {A ∈ B(H) : AT = TA} is called the commutant of T . By a subspace we always mean a closed linear subspace. If A ⊂ B(H...

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp. multi-T...

Let A be a unital Banach algebra and let J be a closed two-sided ideal of A. We prove that if any invertible element of A/J has an invertible lifting in A, then the quotient homomorphism Φ : A → A/J is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foiaş, and Tannenbaum. This yields spectral v...

When is the collection of [Formula: see text]-Toeplitz operators with respect to a tuple commuting bounded text], which has symmetrized polydisc as spectral set, nontrivial? The answer in terms powers text] well unitary extension. En route, Brown–Halmos relations are investigated. A commutant lifting theorem established. Finally, we establish general result connecting text]-algebra generated by...

Identified are a number of conditions on square patterns that are closely related to allowing commutativity with the full pattern. Implications and examples that show non-implications are given, along with a graph that summarizes the provided information. A complete description of commutativity with the full pattern is given in both the irreducible case and the reducible case in which there are...

Let π be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set Bπ of Bessel vectors for π is dense in H, then for any vector x ∈ H the analysis operator Θx makes sense as a densely defined operator from Bπ to ` (G)-space. Two vectors x and y are called π-orthogonal if the range spaces of Θx and Θy are orthogonal, and they are π weakly equivalent i...

In this paper we consider two von Neumann subalgebras B0 and B of a type II1 factor N . For a map φ on N , we define ‖φ‖∞,2 = sup{‖φ(x)‖2 : ‖x‖ ≤ 1}, and we measure the distance between B0 and B by the quantity ‖EB0 −EB‖∞,2. Under the hypothesis that the relative commutant in N of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially is...

We investigate a class of brickwork-like quantum circuits on chains $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these generate unitary dynamics not only when propagating in time direction, but also space direction. consider space-time homogeneous (Floquet) and perturb them with quenched single-site disorder, i.e. by applying independent single site ran...