We introduce the basic definitions and notation that we have used in this book. A basic introduction to operator theory is given in Gohberg and Goldberg (1981), covering most of what is needed for the development of Chapter 22. A much more detailed account is given in Simon (2015). Let

\((\mathsf {H},\Vert \cdot \Vert _{\mathsf {H}})\) and

\((\mathsf {G},\Vert \cdot \Vert _{\mathsf {G}})\) be complex Banach spaces. Whenever there is no ambiguity on the space, we write

\(\Vert \cdot \Vert \) instead of

\(\Vert \cdot \Vert _{\mathsf {H}}\). A function

\(A: \mathsf {H}\rightarrow \mathsf {G}\) is a linear operator from

\(\mathsf {H}\) to

\(\mathsf {G}\) if for all

\(x, y \in \mathsf {H}\) and

\(\alpha \in \mathbb {C}\),

\(A(x+y)= A(x) + A(y)\) and

\(A(\alpha x)= \alpha A(x)\). For convenience, we often write

*Ax* instead of

*A*(

*x*). The linear operator

*A* is said to be bounded if

\(\sup _{\Vert x\Vert _{\mathsf {H}} {\le } 1} \Vert Ax \Vert _{\mathsf {G}} {<} \infty \). The (operator) norm of

*A*, written

Open image in new window , is given by

The identity operator

\(\mathrm {I}: \mathsf {H}\rightarrow \mathsf {H}\), defined by

\(\mathrm {I}x= x\), is a bounded linear operator, and its norm is 1. Denote by

\(\mathsf {BL}(\mathsf {H},\mathsf {G})\) the set of bounded linear operators from

\(\mathsf {H}\) to

\(\mathsf {G}\). For simplicity,

\(\mathsf {BL}(\mathsf {H},\mathsf {H})\) will be abbreviated

\(\mathsf {BL}(\mathsf {H})\). If

\(A \in \mathsf {BL}(\mathsf {H})\), we will use the shorthand notation

Open image in new window instead of

Open image in new window . If

*A* and

*B* are in

\(\mathsf {BL}(\mathsf {H},\mathsf {G})\), it is easy to check that

- (i)
\(\alpha A + \beta B \in \mathsf {BL}(\mathsf {H},\mathsf {G})\), for all \(\alpha , \beta \in \mathbb {C}\);

- (ii)
Open image in new window , for all \(\alpha \in \mathbb {C}\);

- (iii)

- (iv)
if \(A, B \in \mathsf {BL}(\mathsf {H})\) then defining *AB* by \(AB x= A(Bx)\), we have \(AB, BA \in \mathsf {BL}(\mathsf {H})\) and Open image in new window .

An operator \(A \in \mathsf {BL}(\mathsf {H})\) is called invertible if there exists an operator \(A^{-1} \in \mathsf {BL}(\mathsf {H})\) such that \(A A^{-1} x= A^{-1} A x\) for every \(x \in \mathsf {H}\). The operator \(A^{-1}\) is called the inverse of *A*. If *A* and *B* are invertible operators in Open image in new window , then *AB* is invertible and \((AB)^{-1}= B^{-1} A^{-1}\).

The kernel of \(A \in \mathsf {BL}(\mathsf {H})\) is denoted by Open image in new window . It is the closed subspace defined by \(\left\{ x \in \mathsf {H}\,:\;Ax=0\right\} \). The operator *A* is said to be injective if Open image in new window . The *range* (or *image*) of *A*, written Open image in new window , is the subspace \(\left\{ A x\,:\;x \in \mathsf {H}\right\} \). If Open image in new window is finite-dimensional, *A* is called an operator of *finite rank* and Open image in new window is the rank of *A*.

If we define the distance *d*(*T*, *S*) between the operators *T* and *S* to be Open image in new window , then Corollary 22.A.6 shows that the set \(\mathsf {X}\) of invertible operators in \(\mathsf {BL}(\mathsf {H})\) is an open set in the sense that if *T* is in \(\mathsf {X}\), then there exists \(r > 0\) such that \(d(T, S) < r\) implies \(S \in \mathsf {X}\).

Also, the inverse operation is continuous with respect to *d*, i.e., if *T* is invertible and \(d(T , T_n) \rightarrow 0\), then \(T_n\) is invertible for all *n* sufficiently large and \(d(T^{-1}, T_n^{-1} ) {\rightarrow } 0\).

Given a linear operator *T* that maps a finite-dimensional vector space \(\mathsf {H}\) into \(\mathsf {H}\), it is well known from linear algebra that the equation \(\lambda x - Tx = y\) has a unique solution for every \(y \in \mathsf {H}\) if and only if \(\det (\lambda \mathrm {I}-T) \ne 0\), where by abuse of notation, *T* is the matrix associated to the operator *T* in a given basis of \(\mathsf {H}\). Therefore, \(\lambda \mathrm {I}- T\) is invertible for all but a finite number of \(\lambda \). If \(\mathsf {H}\) is an infinite-dimensional Banach space, then the set of those \(\lambda \) for which \(\lambda \mathrm {I}- T\) is not invertible is a set that is usually more difficult to determine.

If

\(\lambda \in {\text {Res}}(T|\mathsf {H})\), then for all

\(x \in \mathsf {H}\),

$$ \Vert (T - \lambda \mathrm {I}) R_\lambda (T) x \Vert _{\mathsf {H}} = \Vert x \Vert _{\mathsf {H}} \;, $$

so if

\(y= R_\lambda (T) x\), then

Open image in new window , which implies

showing that

As a consequence, the spectrum \({\text {Spec}}(T|\mathsf {H})\) is a nonempty compact subset of Open image in new window

A complex number \(\lambda \) is called an eigenvalue of \(T \in \mathsf {BL}(\mathsf {H})\) if there exists \(y \ne 0 \in \mathsf {H}\) such that \(T y = \lambda y\), or equivalently, Open image in new window . The vector *y* is called an eigenvector of *T* corresponding to the eigenvalue \(\lambda \). Every linear operator on a finite-dimensional Euclidean space over \(\mathbb {C}\) has at least one eigenvalue. However, an operator on an infinite-dimensional Banach space may have no eigenvalues.

Let

Open image in new window and

Open image in new window be two Hilbert spaces and

\(T \in \mathsf {BL}(\mathsf {H},\mathsf {G})\). For each

\(y \in \mathsf {G}\), the functional

Open image in new window is a bounded linear functional on

\(\mathsf {H}\). Hence the Riesz representation theorem guarantees the existence of a unique

\(y^* \in \mathsf {H}\) such that for all

\(x \in \mathsf {H}\),

Open image in new window . This gives rise to an operator

\(T^* \in \mathsf {BL}(\mathsf {G},\mathsf {H})\) defined by

\(T^*y = y^*\) satisfying

The operator

\(T^*\) is called the

*adjoint* of

*T*.

Now we will consider the case in which

\(\mathsf {H}, \mathsf {G}\) are Banach spaces and

\(T\in \mathsf {BL}(\mathsf {H},\mathsf {G})\). For

\(\mu \in \mathsf {G}^*\),

\(\nu \in \mathsf {H}^*\), we use the notation

Open image in new window ,

\(x \in \mathsf {G}\),

Open image in new window ,

\(y \in \mathsf {H}\). There exists a unique adjoint

\(T^{*}\in \mathsf {BL}(\mathsf {G}^{*},\mathsf {H}^{*})\) that is defined by an equation that generalizes (

22.A.10) to the setting of Banach spaces: for all

\(\mu \in \mathsf {G}^*\) and

\(x \in \mathsf {H}\),

Note, however, that the adjoint is a map

\(T^{*}:\mathsf {G}^{*}\rightarrow \mathsf {H}^{*}\), whereas

\(T: \mathsf {H}\rightarrow \mathsf {G}\). In particular, in contrast to the Hilbert space case, we cannot consider compositions of

*T* with

\(T^{*}\). We have that

\(\Box \)

A self-adjoint operator on the Hilbert space \(\mathsf {H}\) is positive if for all \(f \in \mathsf {H}\), Open image in new window .