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• W2012028162 abstract "In this paper the following theorem will be proved: A necessary rind sufficient condition that the numerical range of a bounded normal )perator T in a Hilbert space is closed is that the set of extreme points ,f the convex closure of the spectrum of T contains no points of the coninuous spectrum of T. A few terms should be properly defined. The numerical range of a inear bounded operator T in a Hilbert space is defined as the set of ill complex numbers of the form (Tx, x) where x is a unit vector in She space. Clearly the numerical range of T contains the point spec-rum of T. Stone [2] proved that the numerical range of a bounded inear operator T is convex, and that if T is normal then the closure if its numerical range is exactly the convex closure K(T) of the ;pectrum of T. A point X of K(T) is defined as an extreme point of K(T) if no line segment joining any two points of K(T) other than X oontains X. Denote by A the set of extreme points of K(T). It is easy to see that A is contained in the spectrum of T, and that the convex losure of A is exactly K(T). Hence the sufficiency of the condition Follows readily from these facts. The necessity of the condition can be proved by contradiction. Suppose that there exists a point ao in A such that ao lies in the zontinuous spectrum of T. Now the numerical range of T is closed, and hence ao lies in it. So we can find a unit vector x in the space such that ao = (Tx, x). Since T is bounded and normal, it has a resolution of the identity {E,}, and T can be represented as an integral with respect to {EI}, [1], i.e.," @default.
• W2012028162 created "2016-06-24" @default.
• W2012028162 creator A5034289612 @default.
• W2012028162 date "1957-01-01" @default.
• W2012028162 modified "2023-10-14" @default.
• W2012028162 title "A condition that a normal operator have a closed numerical range" @default.
• W2012028162 cites W1714751346 @default.
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• W2012028162 doi "https://doi.org/10.1090/s0002-9939-1957-0094711-2" @default.
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