Approximate proper vectors

Abstract

1. Notations and terminology. Our terminology conforms with that of [2]. The inner product of vectors x and y in a Hilbert space 3C is denoted (x, y). An operator in KC is a continuous linear mapping T: SC->3C. The *-algebra of all operators in 3C is denoted L(3C). A complex number i is a proper value for T if there exists a nonzero vector x such that (T-mI)x = 0; such a vector x is a proper vector for T. A complex number ,u is an approximate proper value for T in case there exists a sequence of vectors x. such that IIxnlj = 1 and I TxnuXnII ->0; equivalently, there does not exist a number e >0 such that (T y) * (T -,IJ) 2 sJ. The spectrum of an operator T, denoted s(T), is the set of all complex numbers y such that T-,uI has no inverse. The approximate point spectrum of T, denoted a(T), is the set of all approximate proper values of T. The point spectrum of T, denoted p(T), is the set of all proper values of T. Evidently p(T) Ca(T) Cs(T). If T is normal, s(T) =a(T) (see [2, Theorem 31.2 ]); if T is Hermitian, a(T) contains a (necessarily real) number a such that IaI =| T|I (see [2, Theorem 34.2]), and in particular one has an elementary proof of the fact that the spectrum of T is nonempty.