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2018 | OriginalPaper | Chapter

10. Spectral Theorems

Author : Piotr Sołtan

Published in: A Primer on Hilbert Space Operators

Publisher: Springer International Publishing

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Abstract

Just as in the case of bounded operators, spectral theorem for unbounded operators assumes various forms. We will begin with continuous functional calculus which can be defined exclusively using the z-transform. Next we will move on to Borel functional calculus and finally assign to each self-adjoint operator its spectral measure and discuss functional calculus for unbounded functions.

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previous chapter z-Transform

next chapter Self-Adjoint Extensions of Symmetric Operators

Moreover, one can show that if $\sigma (z_T)\subset \left ]-1,1\right [$ then T is bounded.

Clearly, since ∥ Φ(f _n)∥ < C for some constant C, we also have

$$\displaystyle \begin{aligned} \Phi(f_n)\xi\xrightarrow[n\to\infty]{}\Phi(f)\xi,\qquad \quad \xi\in\mathcal{H}. \end{aligned} $$

Indeed: fix $\xi \in \mathcal {H}$ and for ε > 0 choose ξ′∈D(T) such that $\|\xi -\xi '\|<\tfrac {\varepsilon }{3C}$. Then

$$\displaystyle \begin{aligned} \big\|\Phi(f_n)\xi&-\Phi(f)\xi\big\|\leq\big\|\Phi(f_n)\xi-\Phi(f_n)\xi'\big\|+\big\|\Phi(f_n)\xi'-\Phi(f)\xi'\big\|+\big\|\Phi(f)\xi'-\Phi(f)\xi\big\|\\ &\quad \leq{C}\|\xi-\xi'\|+\big\|\Phi(f_n)\xi'-\Phi(f)\xi'\big\| +C\|\xi-\xi'\|<\tfrac{2\varepsilon}{3M}+\big\|\Phi(f_n)\xi'-\Phi(f)\xi'\big\|. \end{aligned} $$

Therefore, for n large enough, so that $\big \|\Phi (f_n)\xi '-\Phi (f)\xi '\big \|<\tfrac {\varepsilon }{3C}$, the above estimate gives ∥ Φ(f _n)ξ − Φ(f)ξ∥ < ε.

In Sect. 4.2 we recalled the notion of total variation of a measure. Now the variation of a complex measure ν defined on a σ-algebra $\mathfrak {N}$ is the map $|\nu |:\mathfrak {N}\to [0+\infty ]$ defined by

$$\displaystyle \begin{aligned} |\nu|(\Delta)=\sup\bigg\{\sum_{n=1}^N\big|\nu(\Delta_n)\big|\bigg\},\qquad \quad \Delta\in\mathfrak{N} \end{aligned}$$

with the supremum is taken over all measurable partitions of Δ. Then |ν| is a measure and ν is absolutely continuous with respect to |ν|. Also

$$\displaystyle \begin{aligned} \bigg|\int{f}\,d\nu\bigg|\leq\int|f|\,d|\nu| \end{aligned}$$

for any integrable f.

[AkGl]

N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space (Dover Publications, Mineola, 1993)MATH

[Mau]

K. Maurin, Methods of Hilbert Spaces (Polish Scientific Publishers, Warsaw, 1972)MATH

[Ped]

G.K. Pedersen, Analysis Now (Springer, New York, 1995)MATH

[ReSi1]

M. Reed, B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis (Academic Press, London, 1980)

[Rud2]

W. Rudin, Functional Analysis (McGraw-Hill, New York, 1991)MATH

Title: Spectral Theorems
Author: Piotr Sołtan
Publisher: Springer International Publishing
Book: A Primer on Hilbert Space Operators
Print ISBN: 978-3-319-92060-3

Electronic ISBN: 978-3-319-92061-0

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-92061-0_10

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