Some Remarks on the Invariant Subspace Problem for Hyponormal Operators

We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators. 2000 Mathematics Subject Classification. 47B20, 47A15. Let be a separable, infinite-dimensional, complex Hilbert space and denote by ( ) the algebra of all linear and bounded operators on . An operator T ∈ ( ) is called hyponormal (notation: T ∈H( )) if [T∗,T ] := T∗T −TT∗ ≥ 0, or equivalently, if ‖T∗x‖ ≤ ‖Tx‖ for every x ∈ . The purpose of this paper is to use several results that may be applied to the invariant subspace problem (ISP) for hyponormal operators and thus to bring into focus what remains to be done to solve the problem completely. We begin by recalling some standard notation and terminology to be used. For a (nonempty) compact subset K ⊂ C, we denote by C(K) the Banach algebra of all continuous complex-valued functions on K with the supremum norm, by Rat(K) the subalgebra of C(K) consisting of all rational functions with poles off the set K, and by R(K) the closure in C(K) of Rat(K). For T ∈ ( ), the spectrum of T is denoted by σ(T) and the algebra {r(T) : r ∈ Rat(σ(T))} by Rat(T). The rational cyclic multiplicity of T (notation: m(T)) is the smallest cardinal numbermwith the property that there arem vectors {xi}0≤i<m in such that∨{Axi | 0≤ i <m, A∈ Rat(T)} = . For a bounded (nonempty) open subset U ⊂ C, one denotes by H∞(U) the Banach algebra of those analytic complex-valued functions on U with the property that ‖f‖∞,U := supz∈U |f(z)| < ∞. The ideal of all compact operators on will be denoted by K =K( ). Since K is a two-sided, normclosed ideal in ( ), the quotient algebra ( )/K is a C∗-algebra, which is called the Calkin algebra, and the quotient map from ( ) to ( )/K will be denoted by π . For T in ( ), we write σe(T) (resp., σre(T),σle(T)) for the essential (resp., r ight-, left-essential) spectrum of T (i.e., the spectrum (resp., right, left spectrum) of π(T)). An operator A∈K( ) is called a trace-class operator (notation: A∈ 1( )) if the series tr(|A|) := i∈N〈|A|ei,ei〉 is convergent, where |A| = (A∗A)1/2 and {ei}i∈N is an orthonormal basis of . An operator A∈K( ) is Hilbert-Schmidt operator (notation: A ∈ 2( )) if A∗A ∈ 1( ). For a selfadjoint operator A ∈ ( ), A− will denote its negative part (|A|−A)/2. Finally, μ will denote planar Lebesgue measure defined on the Borel subsets of R2.