Solvability of (λ, μ)-Commuting Operator Equations for Bounded Generalization of Hyponormal Operators

Abstract

Recently, new generalizations have been presented for the hyponormal operators, which are (N, k)-hyponormal operators and (h, M)-hyponormal operators. Some properties of these concepts have also been proved, one of these properties is that the product of two (N, k)-hyponormal operator is also (N, k)- hyponormal operator and the product of two (h, M)-hyponormal operators is (h, M)-hyponormal operator. In our research, we will reprove these properties by using the (l,m)-commuting operator equations, in addition to that we will solve the (l, m)-commuting operator equations for (N, k)-hyponormal operators and (h, M)-hyponormal operators.

New Generalizations for Ϻ-Hyponormal Operators

“…In 2022, Prasad 𝑇. , et.al [14], given important theorem which Fuglede-Putnam symmetric theory of not bounded Ϻ-hyponormal operators is stated. Moreover, he showed that if 𝑇 define on suitable Hilbert space H which is densely Ϻhyponormal operator, N is subspace closed of suitable Hilbert space H has invariant property, the operators 𝑇 and 𝑇|𝑁 are normal, then N reduces T. In 2022 Mohsen S.D introduced the solutions operator equations of kinds (λ,μ)-Commuting Operator Equations with Generalizations Hyponormal Operators define on suitable Hilbert space H, [15]. This paper, presents some basic definitions that we need in our work, and we define the concept of (Ϻ, θ)-hyponormal operator and we give some properties for (Ϻ, θ)-hyponormal operators and shows some properties that are not realized when the operators are (Ϻ, θ)hyponormal operators.…”

Novel Results of K Quasi (ℷ-M)-hyponormal Operator

“…[1], first articulated the idea of hyponormality in [1] under a different name which is subnormal. Despite this, the later two concepts are not merely easy adaptations of one to another because many attributes do not transfer well when taking a djoints, where are the unilateral shift operator, it is a well-known example of a hyponormal operator, which is crucial, also solvability of the ℷ-commuting operator equation takes the following form 𝒯 1 𝒯 2 =ℷ 𝒯 2 𝒯 1 , is one of the key applications of the hyponormal operator, with the equation for the (ℷ, ℳ)-commuting has formulations 𝒯 1 𝒯 2 =ℷ 𝒯 2 𝒯 1 , and 𝒯 * 1 𝒯 2 =ℳ 𝒯 2 𝒯 * 1 , [2] Putnam C.R. [3] investigated several features of the operator 𝐽 = 𝑒𝑖 𝑇 + 𝑒𝑖 𝑇 * , such that in the case 𝑇 discrebe the hyponormal operator in 1957 and continuous this studies on hyponormal until 1961 several hyponormal operator qualities were presented to Berberain S.K.…”