# Chapter 135. Operator Theory

# Chapter 135. Operator Theory

## 135.1 Introduction

Operator theory studies linear maps acting on vector spaces with additional analytic structure, especially normed spaces, Banach spaces, and Hilbert spaces.

In finite-dimensional linear algebra, a linear transformation is usually represented by a matrix. In infinite-dimensional spaces, many linear transformations still behave like matrices, but new issues appear. A linear operator may fail to be bounded. Its spectrum may contain values that are not eigenvalues. Its inverse may exist algebraically but fail to be continuous. Its domain may be a proper subspace.

Operator theory extends matrix theory to infinite-dimensional settings. It is a central part of functional analysis and appears in differential equations, quantum mechanics, harmonic analysis, numerical analysis, and partial differential equations. Operator theory studies linear operators on function spaces, including differential and integral operators, and depends heavily on the topology of those spaces.

## 135.2 Linear Operators

Let \(V\) and \(W\) be vector spaces over a field \(F\). A linear operator is a linear map

$$
T:V\to W
$$

satisfying

$$
T(u+v)=T(u)+T(v),
$$

and

$$
T(av)=aT(v).
$$

When \(V=W\), one often writes

$$
T:V\to V
$$

and calls \(T\) an operator on \(V\).

In finite dimensions, every linear operator on \(F^n\) is represented by an \(n\times n\) matrix. In infinite dimensions, one often studies operators on spaces of sequences, functions, or distributions.

Examples include:

| Operator | Formula | Space |
|---|---|---|
| Shift | \(S(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)\) | Sequence spaces |
| Differentiation | \(Df=f'\) | Function spaces |
| Multiplication | \(M_gf=gf\) | Function spaces |
| Integral operator | \(Tf(x)=\int K(x,y)f(y)\,dy\) | Function spaces |
| Projection | \(Pv\) equals component in a subspace | Hilbert spaces |

These operators generalize matrices, but their analytic behavior can be much richer.

## 135.3 Normed Spaces

A normed vector space is a vector space \(X\) equipped with a norm

$$
\|\cdot\|:X\to\mathbb{R}_{\ge0}.
$$

The norm satisfies:

| Property | Formula |
|---|---|
| Positivity | \(\|x\|\ge0\), and \(\|x\|=0\) only if \(x=0\) |
| Homogeneity | \(\|ax\|=|a|\|x\|\) |
| Triangle inequality | \(\|x+y\|\le \|x\|+\|y\|\) |

The norm introduces size, distance, convergence, and continuity.

Operator theory depends on this added structure. A linear map may preserve algebraic operations while behaving badly with respect to limits. Thus continuity becomes a central issue.

## 135.4 Bounded Operators

Let \(X\) and \(Y\) be normed vector spaces. A linear operator

$$
T:X\to Y
$$

is bounded if there exists a constant \(C\ge0\) such that

$$
\|Tx\|\le C\|x\|
$$

for all

$$
x\in X.
$$

The least such \(C\) is the operator norm:

$$
\|T\| =
\sup_{\|x\|\le1}\|Tx\|.
$$

A bounded linear operator sends bounded sets to bounded sets. In finite-dimensional spaces, every linear map is bounded. In infinite-dimensional spaces, this is no longer automatic.

Boundedness is equivalent to continuity for linear maps between normed spaces. This is one of the first bridges between algebra and analysis.

## 135.5 The Space of Bounded Operators

The set of all bounded linear operators from \(X\) to \(Y\) is denoted

$$
\mathcal{B}(X,Y).
$$

When \(X=Y\), one writes

$$
\mathcal{B}(X).
$$

With addition, scalar multiplication, composition, and the operator norm, \(\mathcal{B}(X)\) behaves like an algebra of infinite-dimensional matrices.

If \(X\) is a Banach space, then \(\mathcal{B}(X)\) is also a Banach space.

This fact allows one to study limits of operators using analytic methods.

## 135.6 Banach Spaces

A Banach space is a complete normed vector space.

Completeness means that every Cauchy sequence converges to an element inside the space.

Examples include:

| Space | Description |
|---|---|
| \(\ell^p\) | \(p\)-summable sequences |
| \(C([a,b])\) | Continuous functions on a compact interval |
| \(L^p(\Omega)\) | \(p\)-integrable functions |
| \(c_0\) | Sequences converging to zero |

Banach spaces are the natural setting for bounded operator theory.

Many major results of functional analysis, including the open mapping theorem, the closed graph theorem, and the uniform boundedness principle, concern operators between Banach spaces.

## 135.7 Hilbert Spaces

A Hilbert space is a complete inner product space.

Its inner product is written

$$
\langle x,y\rangle.
$$

The norm is induced by the inner product:

$$
\|x\|=\sqrt{\langle x,x\rangle}.
$$

Hilbert spaces are especially close to Euclidean spaces. They have orthogonality, projections, adjoints, and spectral theory.

Examples include:

| Hilbert space | Description |
|---|---|
| \(\mathbb{R}^n\) | Finite-dimensional Euclidean space |
| \(\mathbb{C}^n\) | Finite-dimensional complex Hilbert space |
| \(\ell^2\) | Square-summable sequences |
| \(L^2(\Omega)\) | Square-integrable functions |

Most of the spectral theory used in quantum mechanics and harmonic analysis is Hilbert-space operator theory.

## 135.8 Adjoints

Let \(H\) be a Hilbert space and let

$$
T:H\to H
$$

be a bounded linear operator.

The adjoint of \(T\) is the unique bounded operator

$$
T^*:H\to H
$$

satisfying

$$
\langle Tx,y\rangle =
\langle x,T^*y\rangle
$$

for all

$$
x,y\in H.
$$

In finite-dimensional complex spaces, the adjoint is the conjugate transpose of a matrix:

$$
T^*=\overline{T}^{\,T}.
$$

Adjoints are central because they allow geometric structure to interact with operator algebra.

## 135.9 Self-Adjoint Operators

An operator \(T\) on a Hilbert space is self-adjoint if

$$
T=T^*.
$$

Self-adjoint operators generalize real symmetric matrices and Hermitian matrices.

They satisfy many strong properties:

| Property | Meaning |
|---|---|
| Real spectrum | Spectral values lie on the real line |
| Orthogonal spectral decomposition | Generalized diagonalization |
| Variational principles | Eigenvalues described by optimization |
| Physical observables | Quantum-mechanical interpretation |

In finite dimensions, every self-adjoint matrix is unitarily diagonalizable.

In infinite dimensions, diagonalization becomes spectral representation.

## 135.10 Normal Operators

An operator \(T\) is normal if

$$
TT^*=T^*T.
$$

This class includes:

| Operator type | Condition |
|---|---|
| Self-adjoint | \(T=T^*\) |
| Unitary | \(T^*T=TT^*=I\) |
| Skew-adjoint | \(T^*=-T\) |

Normal operators are the correct infinite-dimensional analogue of matrices that can be diagonalized by an orthonormal basis.

The spectral theorem for normal operators is one of the central theorems of operator theory. In broad terms, the spectral theorem identifies operators that can be modeled by multiplication operators, which are the infinite-dimensional analogue of diagonal matrices.

## 135.11 Projections

A projection is an operator \(P\) satisfying

$$
P^2=P.
$$

If \(H\) is a Hilbert space, an orthogonal projection also satisfies

$$
P=P^*.
$$

Orthogonal projections generalize the familiar operation of projecting a vector onto a subspace.

If \(M\subseteq H\) is a closed subspace, then every vector \(x\in H\) has a unique decomposition

$$
x=m+n,
$$

where

$$
m\in M,
\qquad
n\in M^\perp.
$$

The projection sends

$$
x\mapsto m.
$$

Projection operators encode geometric decomposition.

## 135.12 Compact Operators

A bounded operator \(T:X\to Y\) is compact if it sends bounded sets to relatively compact sets.

In practical terms, compact operators behave like limits of finite-rank operators.

Compact operators are important because their spectral behavior resembles finite-dimensional matrix theory more closely than general bounded operators.

For compact operators on infinite-dimensional Hilbert spaces, nonzero spectral values often behave like eigenvalues with finite-dimensional eigenspaces, and they may accumulate only at zero.

Integral operators with sufficiently regular kernels often define compact operators.

## 135.13 Spectrum

Let \(T\in\mathcal{B}(X)\), where \(X\) is a complex Banach space.

The resolvent set of \(T\) is the set of complex numbers \(\lambda\) such that

$$
T-\lambda I
$$

is invertible and has a bounded inverse.

The spectrum is the complement of the resolvent set:

$$
\sigma(T) =
\{\lambda\in\mathbb{C}:T-\lambda I \text{ is not boundedly invertible}\}.
$$

In finite dimensions, the spectrum is exactly the set of eigenvalues.

In infinite dimensions, the spectrum may contain values that are not eigenvalues.

This is one of the main differences between matrix theory and operator theory.

## 135.14 Point, Continuous, and Residual Spectrum

The spectrum is often divided into parts.

| Spectral part | Meaning |
|---|---|
| Point spectrum | \(T-\lambda I\) is not injective |
| Continuous spectrum | Inverse exists on a dense range but is unbounded |
| Residual spectrum | Range is not dense |

The point spectrum is the set of eigenvalues.

The continuous spectrum has no finite-dimensional analogue in ordinary matrix theory. It appears naturally in differential operators and multiplication operators.

For example, multiplication by \(x\) on \(L^2([0,1])\) has spectrum \([0,1]\), but most spectral values are not eigenvalues.

## 135.15 Resolvent

The resolvent of \(T\) is the operator-valued function

$$
R(\lambda,T) =
(T-\lambda I)^{-1}
$$

defined for

$$
\lambda\notin\sigma(T).
$$

The resolvent studies how the inverse changes as the parameter \(\lambda\) varies.

It is a central tool because spectral information is encoded in the analytic behavior of the resolvent.

Many parts of operator theory can be phrased as estimates on

$$
\|(T-\lambda I)^{-1}\|.
$$

For differential operators, resolvent estimates often imply regularity, stability, and decay properties.

## 135.16 Spectral Radius

The spectral radius of a bounded operator is

$$
r(T) =
\sup\{|\lambda|:\lambda\in\sigma(T)\}.
$$

For bounded operators on complex Banach spaces, it is related to powers of the operator by the spectral radius formula:

$$
r(T) =
\lim_{n\to\infty}\|T^n\|^{1/n}.
$$

This formula generalizes matrix spectral radius theory.

It connects long-term iteration behavior with spectral structure.

For example, if

$$
r(T)<1,
$$

then the powers

$$
T^n
$$

tend to zero in many important settings.

## 135.17 Unbounded Operators

Many important operators are not bounded.

The derivative operator

$$
D f=f'
$$

on a function space is a typical example. Its size can grow without bound relative to the size of \(f\).

Unbounded operators cannot be defined on the whole Banach space in the same way as bounded operators. They usually have a domain

$$
\mathcal{D}(T)\subseteq X.
$$

Thus an unbounded operator is written

$$
T:\mathcal{D}(T)\to X.
$$

The domain is part of the operator.

This is essential. Two differential operators with the same formula but different domains may have different spectra.

## 135.18 Closed Operators

An unbounded operator \(T:\mathcal{D}(T)\to X\) is closed if its graph is closed in

$$
X\times X.
$$

That means whenever

$$
x_n\to x
$$

and

$$
Tx_n\to y,
$$

with

$$
x_n\in\mathcal{D}(T),
$$

then

$$
x\in\mathcal{D}(T)
$$

and

$$
Tx=y.
$$

Closedness is a substitute for boundedness in many parts of unbounded operator theory.

Many differential operators are studied as closed operators.

## 135.19 Differential Operators

Differential operators are among the main sources of operator theory.

Examples include:

| Operator | Formula |
|---|---|
| First derivative | \(D=\frac{d}{dx}\) |
| Laplacian | \(\Delta=\sum_i \frac{\partial^2}{\partial x_i^2}\) |
| Schrödinger operator | \(-\Delta+V\) |
| Heat operator | \(\partial_t-\Delta\) |

These operators act on function spaces.

Their spectral properties encode analytic and physical information.

For example, eigenvalues of the Laplacian describe vibration modes, heat diffusion, and geometric structure.

## 135.20 Multiplication Operators

Let \(g\) be a function, and define

$$
(M_gf)(x)=g(x)f(x).
$$

This is a multiplication operator.

Multiplication operators are important because they are the infinite-dimensional analogue of diagonal matrices.

For a diagonal matrix, each coordinate is multiplied by a scalar. For a multiplication operator, each point \(x\) is multiplied by the scalar \(g(x)\).

The spectral theorem says, in broad form, that many normal operators can be represented as multiplication operators.

This is why multiplication operators are a basic model in operator theory.

## 135.21 Shift Operators

The unilateral shift on \(\ell^2\) is defined by

$$
S(x_1,x_2,x_3,\ldots) =
(0,x_1,x_2,x_3,\ldots).
$$

It is an isometry, since

$$
\|Sx\|=\|x\|.
$$

But it is not unitary, because it is not onto.

The shift operator is one of the simplest infinite-dimensional operators whose behavior differs from finite-dimensional intuition.

It plays a central role in Hardy spaces, dilation theory, and operator models.

A classification result known as the Wold decomposition says that every isometry on a Hilbert space decomposes into a unitary part and shift-like parts.

## 135.22 Operator Algebras

A collection of operators may form an algebra.

An operator algebra is an algebra whose elements are operators and whose multiplication is composition.

Important examples include:

| Algebra | Description |
|---|---|
| \(\mathcal{B}(H)\) | All bounded operators on a Hilbert space |
| \(C^*\)-algebra | Norm-closed algebra with adjoint operation |
| von Neumann algebra | Operator algebra closed in weak operator topology |
| Compact operators | Norm-closed ideal in \(\mathcal{B}(H)\) |

Operator algebras connect linear algebra, topology, analysis, and quantum theory.

The theory of operator algebras studies algebras such as \(C^*\)-algebras and is a major part of operator theory.

## 135.23 Functional Calculus

Functional calculus assigns functions of an operator.

For a diagonalizable matrix

$$
A=V D V^{-1},
$$

one defines

$$
f(A)=V f(D) V^{-1},
$$

where \(f(D)\) applies \(f\) to the diagonal entries.

Operator theory generalizes this idea.

For suitable operators, one may define:

$$
e^T,\qquad
\sqrt{T},\qquad
(T-\lambda I)^{-1},\qquad
\log T.
$$

Functional calculus is essential in spectral theory, semigroup theory, differential equations, and quantum mechanics.

## 135.24 Semigroups of Operators

A one-parameter semigroup of operators is a family

$$
(T(t))_{t\ge0}
$$

satisfying

$$
T(0)=I,
$$

and

$$
T(t+s)=T(t)T(s).
$$

Such semigroups describe time evolution.

For example, the heat equation can be written abstractly as

$$
u'(t)=Au(t),
$$

with solution

$$
u(t)=T(t)u(0).
$$

The operator \(A\) is called the generator of the semigroup.

Sectorial operators are important in this context. They are operators whose spectrum lies in a sector and whose resolvent satisfies suitable bounds; they occur in the theory of elliptic and parabolic partial differential equations and as generators of analytic semigroups.

## 135.25 Operator Theory and Linear Algebra

Operator theory extends linear algebra in several directions.

| Linear algebra | Operator theory |
|---|---|
| Matrix | Linear operator |
| Finite-dimensional vector space | Banach or Hilbert space |
| Matrix norm | Operator norm |
| Transpose or conjugate transpose | Adjoint |
| Eigenvalues | Spectrum |
| Diagonalization | Spectral theorem |
| Matrix inverse | Bounded inverse and resolvent |
| Orthogonal projection | Projection onto closed subspace |
| Matrix algebra | Operator algebra |

The main shift is from finite-dimensional algebra to infinite-dimensional algebra plus topology.

## 135.26 Summary

Operator theory studies linear transformations on spaces with analytic structure.

The central ideas are:

| Concept | Meaning |
|---|---|
| Bounded operator | Linear operator controlled by a norm |
| Operator norm | Size of an operator |
| Banach space | Complete normed vector space |
| Hilbert space | Complete inner product space |
| Adjoint | Operator defined by inner product duality |
| Self-adjoint operator | Infinite-dimensional Hermitian analogue |
| Normal operator | Operator satisfying \(TT^*=T^*T\) |
| Compact operator | Infinite-dimensional analogue of finite-rank behavior |
| Spectrum | Generalized eigenvalue set |
| Resolvent | Inverse family \((T-\lambda I)^{-1}\) |
| Unbounded operator | Operator with proper domain |
| Closed operator | Operator with closed graph |
| Operator algebra | Algebra of operators under composition |
| Functional calculus | Method for defining functions of operators |

Operator theory begins with linear algebra but adds norm, topology, convergence, and domain. It explains how matrices generalize to function spaces and why infinite-dimensional linear maps require analytic methods.