Rubén A. Martínez-Avendaño

When do Toeplitz and Hankel operators commute

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolzψ, where ψ is the symbol of the Toeplitz operator, and ψ is an affine function of the characteristic function of certain “anti-symmetric” sets of the unit circle.

Hypercyclicity of shifts on weighted Lp spaces of directed trees

In this paper, we study the hypercyclicity of forward and backward shifts on weighted L spaces of a directed tree. In the forward case, only the trivial trees may support hypercyclic shifts, in which case the classical results of Salas [21] apply. For the backward case, nontrivial trees may support hypercyclic shifts. We obtain necessary conditions and sufficient conditions for hypercyclicity of the backward shift and, in the case of a rooted tree on an unweighted space, we show that these conditions coincide. In memory of Jaime Cruz Sampedro, mathematician, teacher, colleague, and friend.

Essentially Hankel Operators

We introduce the set of essentially Hankel operators and investigate some of its properties. We show in particular that the set contains some operators not of the form “Hankel plus compact”, even when we restrict ourselves to the class of essentially Hankel operators with trivial (Fredholm) index function.

Inner products on the space of complex square matrices

Abstract In this paper we study the problem of finding explicit expressions for inner products on the space of complex square matrices M n ( C ) . We show that, given an inner product 〈 ⋅ , ⋅ 〉 on M n ( C ) , with some conditions, there exist positive matrices A j and B j ∈ M n ( C ) , for j = 1 , 2 , … , m such that 〈 X , Y 〉 = ∑ j = 1 m trace ( Y ⁎ A j X B j ) , for all X , Y ∈ M n ( C ) . However, we show that the result does not hold for all inner products. In fact, if the above expression does not hold, we show that there exist positive matrices A j and B j ∈ M n ( C ) , for j = 1 , 2 , … , m such that 〈 X , Y 〉 = − trace ( Y ⁎ A 1 X B 1 ) + ∑ j = 2 m trace ( Y ⁎ A j X B j ) , for all X , Y ∈ M n ( C ) .