Spectral Theory of Linear Operators: And Spectral Systems in by Vladimir MÃ¼ller PDF

# Spectral Theory of Linear Operators: And Spectral Systems in by Vladimir MÃ¼ller PDF

The monograph should still charm either to scholars who wish to know about spectral thought and to specialists within the box. it may well additionally function a reference ebook.

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Extra info for Spectral Theory of Linear Operators: And Spectral Systems in Banach Algebras

Example text

Xn ) = (ϕ(x1 ), . . , ϕ(xn )) : ϕ ∈ cor A . Another important closed subset of M(A) is the Shilov boundary Γ(A). We deﬁne the Shilov spectrum σΓ for (x1 , . . , xn ) ∈ An by σΓ (x1 , . . , xn ) = (ϕ(x1 ), . . , ϕ(xn )) : ϕ ∈ Γ(A) . It is easy to see that σΓ has similar properties as the spectrum σ and the approximate point spectrum τ . Later we formulate a result of this kind for general spectral systems. Theorem 16. Let x = (x1 , . . , xn ) be an n-tuple of elements of a commutative Banach algebra A.

Hence the function g can be extended to an entire function (denoted also by g) satisfying a = (x − z)g(z) (z ∈ C). For |z| > x we have g(z) = (x − z)−1 a. Thus g(z) → 0 as z → ∞. By the Liouville theorem, g = 0, and so a = (x − z)g(z) = 0. Let A be a commutative Banach algebra and let U be an open subset of the complex plane. Denote by H ∞ (U, A) the algebra of all bounded analytic functions f : U → A with the norm f U = sup f (z) : z ∈ U . Clearly, H ∞ (U, A) is a commutative Banach algebra. Lemma 7.

To prove (ii), suppose ﬁrst that A is a commutative Banach algebra. Then σ(xy) = ϕ(xy) : ϕ ∈ M(A) = ϕ(x)ϕ(y) : ϕ ∈ M(A) ⊂ ϕ(x)ψ(y) : ϕ, ψ ∈ M(A) = σ(x) · σ(y). In the same way, σ(x + y) ⊂ σ(x) + σ(y). In general, write A0 = {x, y} . 24, A0 is a commutative Banach algebra and it is easy to check that σ A0 (x) = σ A (x), σ A0 (y) = σ A (y), σ A0 (xy) = σ A (xy) and σ A0 (x+y) = σ A (x+y). Thus the result follows from the corresponding inclusions for the commutative Banach algebra A0 . 2. Commutative Banach algebras 21 The following result characterizes multiplicative functionals.