Topological Properties of Paranormal Operators on Hilbert Space

Let B(H) be the set of all bounded endomorphisms (operators) on the complex Hubert space H. T € B(H) is paranormal if \\(T — z/) || = l/d(z, cr(T)) for all z £ cr(T) where d(z, cr(T)) is the distance from z to cr(T), the spectrum of T. If J is the set of all paranormal operators on H, then J contains the normal operators, JÍ, and the hyponormal operators; and J is contained in JL, the set of all T £ B(H) such that the convex hull of cr(T) equals the closure of the numerical range of T. Thus, JÍ C J C i_C B(H). Give B(H) the norm topology. The main results in this paper are (1)J(, J, and X, are nowhere dense subsets of B(H) when dim H > 2, (2) Jl, J, and ¿L are arcwise connected and closed, and (3) Jl is a nowhere dense subset of J when dim/Y Paranormal operators have received considerable attention in the current literature ([16], [17], [19], [20], [21], [23], [24], [25])However, only the various spectral properties of paranormal operators have been discussed. In this paper, the topological properties of the set of all paranormal operators Jon a Hilbert space H aie investigated. We begin by giving the notation to be used and by defining some of the more specialized terminology. The point spectrum and approximate point spectrum of an operator T ate denoted by o AT) and o (T), respectively, z € oAT) is a normal eigenvalue if ix e H: (T zl)x = 0\ = \x € H: (T zl)*x = 0i where / denotes the identity operator on H. zea (T) is a normal approximate eigenvalue of T when (1) \\xj = 1 and ||(r-zi")xJ — 0 as 72^oc imply ||(T zl)*xj — 0 as n^oc, and (2) \\yj = 1 and \\(T zl)*yj -* 0 as 72 — ~ imply \\(T zl)yj -, 0 as 72 —» oo. If 5 is a set of complex numbers, then dS denotes the boundary of S and co(S) denotes the convex hull of S. Let W(T) = closure \(Tx, x): x € H, \\x\\ = 1\ denote the closure of the numerical range of T. Let R(T, z) = (T zl)~ fot each z €p(T), the resolvent set of T. The spectral radius of T is denoted by R (T). T e B (H) is hyponormal if T*T TT* > 0. I. Elementary properties of paranormal operators. Stampfli [19] has shown that every hyponormal operator is paranormal. Therefore, since every normal Presented to the Society, August 28, 1970; received by the editors April 16, 1970. AMS (MOS) subject classifications (1969). Primary 4740.