On λ-commuting operators

Abstract

Abstract. For a scalar λ, two operators T and S are said to λ-commute if T S = λST . In this note we explore the pervasiveness of the operators that λ-commute with a compact operator by characterizing the closure and the interior of the set of operators with this property.For a scalar λ, two operators T and S are said to λ-commute if T S = λST . This notion has received attention in the past. In particular, both [B] and [CC] examined the concept. [B] shows that if T λ-commutes with a compact operator, then T has a non-trivial hyperinvariant subspace. In [CC] it is shown that for an integer n, the commutants of T and T n are different if and only if there is a non-zero operator Y and an nth root of unity, λ = 1, such that T Y = λY T . In [L] some additional properties and examples of λ-commuting operators are explored. In this note we explore the pervasiveness of the operators that λ-commute with a compact operator by characterizing the closure and the interior of the set of operators with this property.Throughout we will denote by B(H) the set of all operators on the Hilbert space H, and by B 0 (H) and B 00 (H), respectively, the set of all compact and finite rank operators on H. For a complex number λ define the following classes of operators:Note that when λ = 1, these classes and their topological properties were examined in [CP]. If the scalar λ is understood, write C 1 , C 2 , and C 3 .It is clear that C 1 ⊂ C 2 ⊂ C 3 and that all three sets are invariant under similarities. Every operator in C 1 has a finite-dimensional invariant subspace

ON λ-Commuting Operators

“…• In [11] authors have proved that if an operator in B(H) λ-commutes with a compact operator, then this operator has a non-trivial hyperinvariant subspace • In [8] Conway and Prajitura characterized the closure and the interior of the set of operators that λ-commute with a compact operator • In [18] Zhang ,Ohawada and Cho have studied the properties of operators that λ-commute with a paranormal operator The aim of this paper is to study the situation for binormal, M -hyponormal, quasi * -paranormal operators. Again other related results are also given.…”

On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

“…Moreover, in [29,31] a description of all nonzero solutions X of (1.8) with λ ∈ (0, +∞) was obtained. Recently, equation (1.8), and even more general ones with a bounded A in place of J α , has attracted attention of several mathematicians (see, for instance, [5,6,26], and [8,10,45]). In particular, some results from [29] on equation (1.8) were rediscovered in [5] and [26] (the case α = 1) and in [6] (the case α ∈ Z + \ {0}).…”

On the Spectral Analysis of Direct Sums of Riemann-Liouville Operators in Sobolev Spaces of Vector Functions

“…Moreover, in [29,31] a description of all nonzero solutions X of (1.8) with λ ∈ (0, +∞) was obtained. Recently, equation (1.8), and even more general ones with a bounded A in place of J α , has attracted attention of several mathematicians (see, for instance, [5,6,26], and [8,10,45]). In particular, some results from [29] on equation (1.8) were rediscovered in [5] and [26] (the case α = 1) and in [6] (the case α ∈ Z + \ {0}).…”