Elementary Properties of Hyponormal Operators and Semi-Hyponormal Operators

Tensor products of operators—strong stability and p-hyponormality

“…The class H p is monotonic decreasing on p; i.e., if T P H p , then T P r q for all 0`q p, and we may assume without loss of generality that 0`p`1 2 X (Indeed one may assume, without loss of generality that p 2 Àn , for some integer n b 1.) H 1 2 À Á operators were introduced by Xia (see [22, p238] for the appropriate reference), and H p operators for a general 0`p`1 2 have since been considered by a number of authors (see [1,2,5,6,22] for further references). Although the class of H p operators 0`p`1 2 , is strictly larger than the class of hyponormal operators, H p operators share a large number of properties with hyponormal operators.…”

“…The operator T is said to be p-hyponormal, 0`p 1, if T Ã j j 2p T j j 2p X Let H p denote the class of phyponormal operators (so that H 1 denotes the class of 1-hyponormal, or simply hyponormal, operators). Although H p , 0`p`1, contains H 1 as a proper subclass, H p operators have spectral properties very similar to those of H 1 operators (see [1,2,5,6,7,22], and some of the references cited in these papers, for further information on H p operators). It is shown that A B P H p if and only if AY B P H p .…”

Back to RS-SR spectral theory

“…A 1-hyponormal operator is a hyponormal operator and 1 2 -hyponormal operators are called semi-hyponormal ( [3,29]). The Löwner-Heinz inequality implies that if A is q-hyponormal then it is p-hyponormal for any 0 < p ≤ q.…”

Seminormal systems of operators in Clifford environments

“…We should also mention that the theory of seminormal operators was initially developed for pairs (X, Y ) of self-adjoint operators in L(H) rather than a single operator T ∈ L(H). Assuming that X = Re(T ) and Y = Im(T ), i.e., X is the real part of T and Y is the imaginary part of T , from For a comprehensive historical perspective on the development of the theory of seminormal operators and the relationship with the theory of subnormal operators we refer to the monographs by Putnam [36], Clancey [8], Xia [43], Martin and Putinar [28], and Conway [9]. Our goal is to single out and motivate what we believe to be the most natural counterparts of the previous equations and requirements in a multidimensional setting, i.e., for systems of operators.…”