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

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

By Vladimir Müller

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

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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.

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