Joint Hyponormality Research Articles

α , β -Normal Operators in Several Variables

We consider an extension of the concept of α , β -normal operators in single variable operator to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality.

Spherical Aluthge transform, spherical p and log-hyponormality of commuting pairs of operators

ABSTRACT In this paper, we introduce the notions of the spherical p-hyponormality and the spherical log-hyponormality for commuting pairs of operators. We first prove that for a commuting pair of operators, if is jointly hyponormal, then is spherically p-hyponormal for . Second, we show that for and , a spherically p-hyponormal operator is spherically q-hyponormal. Third, we completely characterize the spherically p-hyponormal 2-variable weighted shifts. Fourth, we show that if is Taylor invertible and spherically p-hyponormal, then is spherically log-hyponormal. Moreover, it is possible to create a striking example regarding spherically log-hyponormal but not p-hyponormal. Furthermore, we show that the spherical log-hyponormality of with a norm condition implies the spherical log-hyponormality for , where is the spherical Aluthge transform of . Finally, in the case of 2-variable weighted shifts, we show that the spherical Aluthge transform does not preserve joint hyponormality, in sharp contrast with the 1-variable case.

Joint Hyponormality of Toeplitz Pairs with Rational Symbols

In this note we consider joint hyponormality of pairs of Toeplitz operators acting on the Hardy space \({H^2(\mathbb T)}\) of the unit circle \({\mathbb T}\). We give answers on some questions which arise from joint hyponormality of Toeplitz pairs with rational symbols.

Erratum to: Joint Hyponormality of Rational Toeplitz Pairs

In Theorem 1 of the article “Joint Hyponormality of Rational Toeplitz Pairs” [1], the condition ‘φ and ψ have a common pole’ has to be assumed. Due to a technical problem, we omitted an assumption in Theorem 1 of [1]. For the sake of completeness, we restate Theorem 1, in its correct form. Theorem 1. Let φ and ψ be rational functions in L∞ which have a common pole. If T = (Tφ, Tψ) is hyponormal then φ− βψ ∈ H for some constant β. The proof of [1, Theorem 1] is correct and complete under the additional assumption ‘φ and ψ have a common pole.’ In turn, Corollary 2 of [1] should also have an additional assumption ‘θ0 and θ2 are not coprime,’ which is equivalent to the condition ‘φ and ψ have a common pole.’ We would remark that Corollary 2 of [1] is still a generalization of the case of trigonometric polynomial symbols because if φ and ψ are trigonometric (non-analytic) polynomials then they have a common pole at z = 0.

Joint Hyponormality of Rational Toeplitz Pairs

We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞. The main result of this article is as follows. If T = (Tϕ, Tψ) is a hyponormal rational Toeplitz pair then ϕ − βψ ∈H2 for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete characterization of hyponormal rational Toeplitz pairs.

Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal

We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.

Joint hyponormality of composition operators with linear fractional symbols

We study joint hyponormality and joint subnormality of ofn-tuples of commuting composition operators with linear fractional symbols, acting on the Hardy spaceH 2. We also consider subnormality ofn-tuples of adjoints of composition operators.

𝑘-hyponormality of weighted shifts

An operator T T is defined to be k k -hyponormal if the operator matrix ( [ T ∗ j , T i ] ) i , j = 1 k \left ( {\left [ {{T^{ * j}},{T^i}} \right ]} \right )_{i,j = 1}^k is positive, where [ A , B ] = A B − B A \left [ {A,B} \right ] = AB - BA . In A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195, we proved that k k -hyponormality is equivalent to a Bram-type condition, namely, that the operator matrix ( T ∗ j T i ) i , j = 0 k \left ( {{T^{ * j}}{T^i}} \right )_{i,j = 0}^k is positive. In this note we prove that for weighted shifts, k k -hyponormality is equivalent to an Embry-type condition, namely, that the operator matrix ( T ∗ i + j T i + j ) i , j = 0 k \left ( {{T^{ * i + j}}{T^{i + j}}} \right )_{i,j = 0}^k is positive. We give an example to show that this latter condition fails even for a rank one perturbation of a weighted shift. For weighted shifts this Embry condition reduces to the positivity of a sequence of ( k + 1 ) × ( k + 1 ) \left ( {k + 1} \right ) \times \left ( {k + 1} \right ) Hankel matrices and we use this reduction to give a new proof of one of the principal results of Curto.

A Note on Joint Hyponormality

We describe certain cones of polynomials in two variables naturally associated to the class(es) of operators $T$ for which the tuple $(T,{T^2}, \ldots ,{T^n})$ is jointly (weakly) hyponormal. As an application we give an example of an operator $T$ such that the tuple $(T,{T^2})$ is jointly but not weakly hyponormal. Further, we show that there exists a polynomially hyponormal operator which is not subnormal if and only if there exists a weighted shift with the same property.

A note on joint hyponormality

We describe certain cones of polynomials in two variables naturally associated to the class(es) of operators T T for which the tuple ( T , T 2 , … , T n ) (T,{T^2}, \ldots ,{T^n}) is jointly (weakly) hyponormal. As an application we give an example of an operator T T such that the tuple ( T , T 2 ) (T,{T^2}) is jointly but not weakly hyponormal. Further, we show that there exists a polynomially hyponormal operator which is not subnormal if and only if there exists a weighted shift with the same property.

On Joint Hyponormality of Operators

A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hilbert space.

On joint hyponormality of operators

A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hilbert space.