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## Über dieses Buch

The book concisely presents the fundamental aspects of the theory of operators on Hilbert spaces. The topics covered include functional calculus and spectral theorems, compact operators, trace class and Hilbert-Schmidt operators, self-adjoint extensions of symmetric operators, and one-parameter groups of operators.

The exposition of the material on unbounded operators is based on a novel tool, called the z-transform, which provides a way to encode full information about unbounded operators in bounded ones, hence making many technical aspects of the theory less involved.

## Inhaltsverzeichnis

### 1. Spectrum of an Operator

Let ℋ $$\mathcal {H}$$ be a Hilbert space. The Banach space B ( ℋ ) $$\mathsf {B}(\mathcal {H})$$ of bounded operators on ℋ $$\mathcal {H}$$ is an algebra over ℂ $$\mathbb {C}$$ with multiplication defined as composition of operators and the identity operator is the unit unit of a C∗-algebra (neutral element of multiplication) of this algebra. The operation of passing to the adjoint operator B ( ℋ ) ∋ x ↦ x ∗ ∈ B ( ℋ ) $$\displaystyle\mathsf {B}(\mathcal {H})\ni {x}\longmapsto {x^*}\in \mathsf {B}(\mathcal {H})$$ is an anti-linear, anti-multiplicative involution involution (for any x ∈ B ( ℋ ) $$x\in \mathsf {B}(\mathcal {H})$$ we have x ∗∗ = x). Moreover, the operator norm is compatible with algebra structure in the sense that ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ , x , y ∈ B ( ℋ ) . $$\displaystyle\|xy\|\leq \|x\|\|y\|,\qquad\quad{x,y}\in \mathsf {B}(\mathcal {H}).$$ In particular B ( ℋ ) $$\mathsf {B}(\mathcal {H})$$ is a Banach algebra Banach algebra .

Piotr Sołtan

### 2. Continuous Functional Calculus

In this chapter we will introduce by far the most important tool of the theory of operators on Hilbert space, namely functional calculus for self-adjoint operators. We begin with slightly more general considerations focused on normal operators which we will revisit later in Chapter 7 .

Piotr Sołtan

### 3. Positive Operators

One of the first and, incidentally, very effective applications of functional calculus is to take square roots of positive operators. In this chapter we will introduce positive operators and related notions of a projection and a partial isometry. We will prove existence and uniqueness of polar decomposition of bounded operators. We will also briefly investigate the partial order on operators defined by the notion of a positive operator. In particular we will show that any bounded monotonically increasing net of self-adjoint operators has a supremum which is also the limit of the net in strong topology.

Piotr Sołtan

### 4. Spectral Theorems and Functional Calculus

One of the most important facts about operators on finite dimensional spaces equipped with a scalar product is the spectral theorem which says that a self-adjoint operator m can be written as a linear combination

Piotr Sołtan

### 5. Compact Operators

In this chapter we will discuss the most fundamental results on compact operators on Hilbert spaces. It is important to stress that analogous theory for operators on Banach spaces is somewhat more complicated and, in particular, requires more considerations of topological nature. The very definition of a compact operator between Banach spaces is significantly different from the one we will propose in the case of operators on a Hilbert space. In the more general framework of Banach spaces the definition of a compact operator amounts to the conclusion of Proposition 5.1. An interested reader will find more information on compact operators in any functional analysis textbook, e.g. in [Rud2, Chapter 4].

Piotr Sołtan

### 6. The Trace

We will now discuss the concept of the trace of an operator. This notion has applications in quantum physics as well as in many problems of pure mathematics. One of them concerns a precise description of the dual space of B 0 ( ℋ ) $$\mathsf {B}_0(\mathcal {H})$$ which we will give in detail. Analysis of the trace as an “unbounded linear functional” on B ( ℋ ) $$\mathsf {B}(\mathcal {H})$$ leads to far reaching generalizations in the theory of operator algebras (under the guise of theory of weights). Our presentation of this topic will be based on the same foundations which underlie those generalizations and can be treated as an introduction to more advanced techniques of the theory of operator algebras.

Piotr Sołtan

### 7. Functional Calculus for Families of Operators

In this chapter we will extend functional calculus to families of commuting self-adjoint operators. As an application of this extension we will be able to introduce in Sect. 7.4 functional calculus for normal operators. This will be the only part of the book in which we will require some results of the theory of Banach algebras, or more specifically, C∗-algebras. These have been gathered in Appendix A.5 .

Piotr Sołtan

### 8. Operators and Their Graphs

Applications of operator theory in other branches of mathematics and in mathematical physics very often involve operators which are not bounded. This poses numerous difficulties whose source is, for the most part, the lack of useful algebraic structure on the set of unbounded operators. Our presentation of the theory of unbounded operators on a Hilbert space will focus on a few select issues and our preferred strategy for dealing with them will be to reduce them to questions about bounded operators. We will begin with some introductory information gathered in Sect. 8.1. In the following chapters we will introduce our key tool which we call the z-transform and later use this tool to extend various versions of the spectral theorem to unbounded self-adjoint operators. The final chapters will be devoted to several classical topics like self-adjoint extensions of symmetric operators and elements of the theory of one-parameter groups of unitary operators.

Piotr Sołtan

### 9. z-Transform

This chapter will be devoted to developing an extremely useful tool for dealing with unbounded operators, namely the so called z-transform. It was introduced in a context much wider than the theory of operators on Hilbert spaces by S.L. Woronowicz (see [Lan, WoNa]). As we already mentioned a couple of times, the z-transform is a way to encode full information about a given closed densely defined operator T in a bounded operator z T. The procedure of passing from T to z T requires several preliminary results which will be presented in Sect. 9.1. As an illustration of the use of the z-transform we will provide a simple proof of existence of polar decomposition of closed operators given in Sect. 9.3.

Piotr Sołtan

### 10. Spectral Theorems

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.

Piotr Sołtan

### 11. Self-Adjoint Extensions of Symmetric Operators

Self-adjointness of certain operators is crucial in many applications (e.g. in quantum physics or in partial differential equations). However, operators which naturally occur in many problems turn out to be symmetric, but not necessarily self-adjoint. It is for that reason that the problem of existence and classification of self-adjoint extensions of symmetric operators was one of the first challenges of the theory of unbounded operators on Hilbert spaces. In this chapter we will present the first results on that topic.

Piotr Sołtan

### 12. One-Parameter Groups of Unitary Operators

One of particularly fruitful applications of the theory of operators on Hilbert spaces is in representation theory of topological groups. In this chapter we will study basic properties of representation theory of the abelian group ℝ $$\mathbb {R}$$ .

Piotr Sołtan

### Backmatter

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