Learn the Basics of Hilbert Spaces and Their Relatives: Operators

 

Operators. The Maze Of Definitions.

We will use the conventions of part I (Basics), which are $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, $z \mapsto \overline{z}$ for the complex conjugate, $\tau$ for transposing matrices or vectors, which we interpret as written in a column if given a basis, and $\dagger$ for the combination of conjugation and transposition, the adjoint matrices. $\mathcal{H},\mathcal{H}_1,\mathcal{H}_2,\ldots$ indicate Hilbert spaces. Their dual spaces are noted by $\mathcal{H}^*$, the orthogonal complements of a subspace $U$ as $U^{\perp}$. Our inner products will be sesquilinear in the first and linear in the second argument. Integrability usually refers to the Lebesgue measure.

In part I, we defined the objects, which functional analysis and large parts of physics deal with. In the following, I will add the morphisms, the natural functions between those objects. As we deal with vector spaces, our functions will be linear functions:
\begin{equation}
\begin{aligned}
T\, : \,\mathcal{H}_1 &\longrightarrow \mathcal{H}_2\\
T(\alpha\psi + \beta\chi) &= \alpha T(\psi) + \beta T(\chi)
\end{aligned}
\end{equation}
The subspace $D(T) \subseteq \mathcal{H}_1$ of points where the function $T$ is defined is called domain of $T$, the subspace $R(T)=T(D(T))\subseteq \mathcal{H}_2$ is called range or image of $T$, and $N(T)=\{\psi \in D(T)\, : \,T\psi=0\}\subseteq \mathcal{H}_1$ the nullspace of $T$. Functions are traditionally called (linear) operators in functional analysis, and (linear) functionals, if they map to the underlying scalar field $\mathbb{F}$. In case $\mathcal{H}_1=\mathcal{H}_2=\mathcal{H}$ we briefly speak of an operator of $\mathcal{H}$. Although presumably historically originated, these distinctions make sense: Hilbert spaces are usually infinite-dimensional, so matrices are not really helpful and the term operator decouples the association linear function – matrix, as well as already indicates their major role (and origin) in physics; and functionals as the elements of dual spaces, which play an important role in functional analysis, is a convenient shorthand term. The fact that the definition of Hilbert spaces doesn’t include any requirement on dimensionality is important here, although they are primarily meant to investigate infinite-dimensional spaces, because it makes functionals a certain kind of operator – $\mathbb{F}$ is a Hilbert space – which we otherwise would have to achieve by artificial constructions or exceptions.

Fourier and Poincaré observed at the end of the nineteenth century the necessity of a transition from finite to infinite as they studied differential equations in thermodynamics.

“This was obviously not the first idea of a “transition from the finite to the infinite”. Without going back to the ideas of the seventeenth century concerning the transition from the calculus of finite differences to the differential calculus, D. Bernoulli had built up the theory of the vibrating string on such a procedure. In 1836, Sturm noted in his work on the differential equation $y´´- q(x)y + \lambda y = 0$ that his ideas were suggested to him by analogous considerations of a system of equations of differences. From about 1880 onward, the need for a new analysis began to be felt from different sides, which instead of the usual functions “functions with infinitely many variables” were in consideration … Even in classical analysis one finds attempts to account for an “operator calculus”, such as the definitions of fractional-order derivatives by Leibniz and Riemann, or the relation $\gamma (-a) = e^{aD}$ which links translations and derivatives, although it was basically used by Lagrange as an expression for the Taylor series.” [12]

Linear Operators. I.

An example shall demonstrate that the world of linear Hilbert space operators is actually a new one, because it mainly deals with topological rather than linear properties. Recall that $L_2(M)$ is the vector space of square integrable, complex valued functions on a Lebesgue measurable subset $M \subseteq \mathbb{R}^n$ (see I. 5.10). Let $f:M \rightarrow \mathbb{C}$ be a measurable function. Then we define the maximal multiplication operator by $f$ in $L_2(M)$ by
\begin{align*}
T\, : \,L_2(M) &\longrightarrow L_2(M)\\
T(\psi) &= f \cdot \psi \; , \;\psi \in D(T) \text{ with }\\
D(T)&=\{\,\psi \in L_2(M)\,: \,f \cdot \psi \in L_2(M) \,\}
\end{align*}
As a first observation, I want to note, that there has to be much more emphasis on the subspace $D(T)$, on which operators are actually defined, than in the realm of linear algebra. Furthermore, it can be shown that $D(T)\subseteq L_2(M)$ is a dense subspace and the following statements are equivalent:
\begin{align*}
&R(T) \subseteq L_2(M) \text{ is dense } \\
&T \text{ is injective, i.e. }T\psi = 0 \Longrightarrow \psi=0 \\
&f(x) \neq 0 \text{ almost everywhere (a.e.) on } M
\end{align*}
If one of these conditions holds, we can define a maximal multiplication operator $T^{-1}$ in an analog way with the function
$$
f_1(x) =
\begin{cases}
f(x)^{-1} & \text{if } x \in M \text{ and }f(x) \neq 0 \\
0 & \text{if } x \in M \text{ and }f(x) = 0
\end{cases}
$$
Another topological aspect is, that isomorphisms are also isometries, and therefeore sometimes called isometric isomorphisms, i.e. they don’t change the norm: $||T(\psi)||_2=||\psi||_1$. In this case it is also required, that $D(T)=\mathcal{H}_1$ and $R(T)=\mathcal{H}_2\,.$ In case $R(T)$ is finite dimensional, $T$ is also called a finite dimensional linear operator.

Bounded Operators.

The possibility of infinite dimensions not only focuses the consideration on topology it forces them. In the case of finite dimensions, everything around linearity, differentiability, and continuity is fine, however, things get more complicated in infinite dimensions. For instance formulas like
$$
|\psi_j(x)|^2 \leq ||(\psi_1(x),\ldots ,\psi_n(x))||^2 \leq \sum_{j=1}^{n} |\psi_j(x)|^2
$$
stop to make sense for $n=\infty$, if $\psi$ is a function taken from an infinite-dimensional vector space. Thus another tool is needed, to handle at least large classes of functions.

A linear operator $T\, : \,\mathcal{H}_1\longrightarrow \mathcal{H}_2$ is called continuous in $\psi \in D(T)$ if for any sequence $(\psi_n)_{n \in \mathbb{N}} \subseteq D(T)$
$$
\lim_{n \to \infty}\psi_n = \psi \, \Longrightarrow \,\lim_{n \to \infty}T\psi_n = T\psi
$$
If $T$ is continuous in all points of $D(T)$, then $T$ itself is called a continuous linear operator. An operator is called bounded operator, if there is a $C \in \mathbb{R}$ such that for all $\psi \in D(T)$
$$
||T\psi|| \,\leq \,C \cdot ||\psi||
$$
Those two seemingly different properties are actually equivalent:
\begin{align*}
T \text{ is continuous }\, \Longleftrightarrow \,T \text{ is continuous in } 0 \,\Longleftrightarrow \,T \text{ is bounded }
\end{align*}
For a bounded linear operator, we have the least upper bound as operator norm
$$
||T|| = \operatorname{inf}\{\,C \geq 0\,: \, ||T\psi||\leq C\cdot ||\psi|| \text{ for all }\psi \in D(T)\,\}
$$
The vector space of all bounded, and everywhere on the Hilbert space $\mathcal{H}_1$ defined linear operators build a Banach space $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ with this norm.

Adjoint. Hermitian. Symmetric. Self-Adjoint.

We want to define similar concepts for linear operators, as we have it for matrices in finite dimensional vector spaces. As we have an inner product in Hilbert spaces, we are now looking for adjointness, i.e. we want to define for an operator $T:\mathcal{H}_1 \rightarrow \mathcal{H}_2$ another operator, the adjoint operator $T\,^*$, which respects the inner products: $\chi \in D(T)\subseteq \mathcal{H}_1 \; , \; \psi \in D(T\,^*)\subseteq \mathcal{H}_2$
\begin{equation}
\begin{aligned}
T\,^*\, &: \,\mathcal{H}_2 \longrightarrow \mathcal{H}_1 \\
\langle \psi , T\chi \rangle_2 &= \langle T\,^*\psi , \chi \rangle_1
\end{aligned}
\end{equation}
Firstly, as there is no risk of confusion, adjoint operators are usually written with an asterisk, which we also use for dual spaces, instead of e.g. $T\,^{adj}$. Secondly, the infinite-dimensional case requires more caution than the matrix case. Roughly said, the formal way to do it, is to define a formally adjoint operator, i.e. any operator which satisfies (2) on the corresponding domains and then to bother with the topological details of those. We require that $T$ is a densely defined linear operator, i.e. $D(T)$ is a dense subset of $\mathcal{H}_1$ which is automatically true if the operator is defined for all elements. A continuation $\hat{T}$ of an operator $T$ is an operator which extends it, i.e. $D(T) \subseteq D(\hat{T})$ and $T\psi = \hat{T}\psi$ for $\psi \in D(T)$. Vice versa is $T$ a restriction of $\hat{T}$. Instead of topological considerations, I want to list a few properties of adjoint operators, which might be of more interest to physicists.

For a densely defined operator, we can define the operator norm (unbounded allowed) as
\begin{equation}
\begin{aligned}
||T|| &= \operatorname{sup}\,\{\,||T\psi||\, : \,\psi \in D(T)\; , \;||\psi||\leq 1 \,\}\\
&= \operatorname{sup}\,\{\,||T\psi||\, : \,\psi \in D(T)\; , \;||\psi|| = 1 \,\}\\
&= \operatorname{sup}\,\{\,||T\psi||\, : \,\psi \in D(T)\; , \;||\psi|| < 1 \,\} \\
& \stackrel{\mathcal{H}_1 = \mathcal{H}_2}{=} \operatorname{sup}\,\{\,|\langle T\psi,\chi \rangle |\, : \, \psi,\chi \in D(T)\; , \;||\psi||=||\chi||=1 \,\}
\end{aligned}
\end{equation}
Let $T:\mathcal{H}_1 \rightarrow \mathcal{H}_2$ be a densely defined operator, then $N(T\,^*)=R(T)^{\perp}$ and if $T\,^*$ is also densely defined, then $T\,^{**}=(T\,^*)^*$ is a continuation of $T$. Furthermore

• $T$ is bounded, if and only if $T\,^* \in \mathcal{B}(\mathcal{H}_2,\mathcal{H}_1)$ is also bounded.
• If $T$ is bounded, then $||T||=||T\,^*||$
• If $T$ is bounded, then $T\,^{**}$ is the unique continuation of $T$ on $\mathcal{H}_1$
• For $T \in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ we have $T\,^{**}=T$

Let’s consider a linear functional $T : \mathcal{H} \longrightarrow\mathbb{F}$. Then there is a unique $\tau \in \mathcal{H}$ such that $T\psi = \langle \tau,\psi \rangle$ for all $\psi \in \mathcal{H}$ and we have for $z\in \mathbb{F}$
\begin{equation*}
\begin{aligned}
\langle T\,^*z,\psi \rangle &= \langle z, \langle \tau,\psi \rangle \rangle \\
&=\overline{z}\cdot \langle \tau,\psi \rangle \\
&= \langle z \cdot \tau,\psi \rangle \\
\end{aligned}
\end{equation*}
and thus
\begin{equation}
T\,^*(z)=z \cdot \tau
\end{equation}
Given bounded operators defined on entire Hilbert spaces $\mathcal{H}_1 \stackrel{T_1,T}{\longrightarrow} \mathcal{H}_2 \stackrel{T_2}{\longrightarrow} \mathcal{H}_3$ we have
\begin{equation}
\begin{aligned}
(z\cdot T)^* &= \overline{z}\cdot T\,^*\\
(T+T_1)^*&= T\,^*+T_1\,^*\\
(T_2T_1)^* &= T_1\,^*\,T_2\,^*
\end{aligned}
\end{equation}
In case $\mathcal{H}_i=\mathcal{H}$ the bounded and defined everywhere on $\mathcal{H}$ linear operators form a Banach algebra $\mathcal{B}(\mathcal{H})$ with $1$, and the duality given by adjointness is an anti- or semilinear anti-homomorphism of this algebra. For an operator $T\in \mathcal{B}(\mathcal{H})$ we get

\begin{equation} T \, T\,^* , T\,^* \, T \in \mathcal{B}(\mathcal{H}) \\ ||T\,T\,^*||=||T\,^*\,T||=||T||^2 \end{equation}

and both $T \, T\,^* , T\,^* \, T$ are self-adjoint.

A linear operator $T\, : \,\mathcal{H} \longrightarrow \mathcal{H}$ is called Hermitian, if he is formally adjoint to itself, i.e.
\begin{equation}
\begin{aligned}
\langle T\psi , \chi \rangle &= \langle \psi , T\chi \rangle \text{ for all }\psi,\chi \in D(T)
\end{aligned}
\end{equation}
An operator on a complex Hilbert space is Hermitian, if and only if the quadratic form
$$
q(\psi) := \langle \psi,T\psi \rangle
$$
on $D(T)$ is real.

$T$ is called symmetric, if it is Hermitian and densely defined, i.e. $\overline{D(T)}=\mathcal{H}$

$T$ is called self-adjoint, if it is Hermitian, densely defined and $T=T\,^*$.

I now pay the price for my sloppiness with formally adjoint operators, as it looks as if the last definition is the same as the first. Well, basically, yes, but Hermitian only needs some formally adjoint operator, e.g. the one with $D(T)=\{0\}$, and self-adjoint a maximal one among those: $T\,^*$ is formally adjoint to $T$, and a continuation of all formally adjoint operators of $T$. Hermitian thus means we can switch the position of the linear operator in the inner product where it is defined, symmetric means the same plus a dense domain, and self-adjoint means symmetric plus the operator is already the unique continuation of all formally adjoined operators. Fortunately, we have

For bounded linear operators defined everywhere on a Hilbert space, the terms Hermitian, symmetric and self-adjoint are equivalent.

which again stresses the role of bounded operators. A densely defined operator on a Hilbert space is called normal, if $D(T\,^*)=D(T)$ and $||T\psi||=||T\,^*\psi||$ for all $\psi \in D(T).$ A self adjoint operator is normal, and if it is also an injective operator $T$, the inverse operator $T^{-1}$ is also self-adjoint. When dealing with operators, the inverses are defined on $D(T^{-1}) = R(T)$, i.e. not automatically on the entire spaces!

Closed. Closable. Closure. Core.

A bounded operator is closed if and only if its domain $D(T)$ is closed. The operators we’re going to define here, are not necessarily bounded. It’s a bit like the next best property. A linear operator $T$ is closed, if its graph $G(T)=\{\,(\psi, T\psi)\,\vert \,\psi \in D(T)\,\}$ is a closed set in the product topology. It is called closable, if $\overline{G(T)}$ is a graph, in which case there is a unique operator $\overline{T}$ with $G(\overline{T})=\overline{G(T)}$ which is closed and called the closure of $T$.

Let $T$ be a closed operator, then a subspace $D\subseteq D(T)$ is called essential domain or core, if $T=\overline{T|_D}$. The restriction $T|_D$ is closable and $\overline{G(T|_D)} \subseteq G(T)$. If $T$ is a closable operator, then $D(T)$ is the core of its closure $\overline{T}\,.$

If $T$ is closed, $N(T)$ is also closed. If $T$ is injective, then $T$ is closed if and only if $T^{-1}$ is closed. If $T$ is bounded, then it is closable with $D(\overline{ T })=\overline{D(T)}$ and its closure $\overline{T}$ is identical with its continuation $\hat{T}$ on $\overline{D(T)}$.

If $T\, : \,\mathcal{H}_1 \longrightarrow \mathcal{H}_2$ is a densely defined operator, then $T\,^*$ is closed. $T$ is closable if and only if $T\,^*$ is densely defined and $\overline{T}=T\,^{**}$. Is $T$ closable, then $(\overline{T})^*=T\,^*$.

Every symmetric operator of a Hilbert space is closable and its closure $\overline{T}$ is symmetric, too.

Let $T\, : \,\mathcal{H}_1 \longrightarrow \mathcal{H}_2$ be a linear operator. Then we define on $D(T)$
\begin{equation}
\begin{aligned}
\langle \psi,\chi \rangle_T &= \langle \psi,\chi \rangle + \langle T\psi,T\chi \rangle \\
||\psi||_T &= \left(||\psi||^2 + ||T\psi||^2\right)^{1/2}
\end{aligned}
\end{equation}
an inner product and corresponding norm, the $T-$norm or graph-norm. Then $T$ is closed if and only if $(\,D(T),\langle \; , \;\rangle_T\,)$ is a Hilbert space.

Compact Operators.

A linear operator $T\, : \,\mathcal{H}_1 \longrightarrow \mathcal{H}_2$ is called compact, if every bounded sequence in $D(T)$ contains a subsequence $(\psi_n)_{n \in \mathbb{N}} \subseteq D(T)$ such that $(T\psi_n)_{n \in \mathbb{N}}$ converges. Every compact operator is also bounded, and its closure $\overline{T}$ is compact.

Compact operators have some nice properties. However, we first need more definitions. Although we do not have matrices, We’re still considering linear functions, so one of the most important questions is the one about eigenvalues, eigenvectors, and eigenspaces.

“And, however unbelievable this may seem to us, it took quite a long time until it has been clear to mathematicians, that what the algebraists write as $(I – \lambda T)^{-1}$ for a matrix $T$, is essentially the same as the analysts represent by $I+\lambda T+\lambda^2T^2+\ldots$ for a linear operator $T$“. [12]

Eigen-World.

Let $T\, : \,\mathcal{H} \longrightarrow \mathcal{H}$ be a linear operator on the Hilbert space $\mathcal{H}$. A number $\lambda \in \mathbb{F}$ is called an eigenvalue of $T$, if there is a vector $\psi \in D(T)-\{0\}$ with $T\psi = \lambda \cdot \psi$. This means, that $\lambda -T = \lambda\cdot I- T$ has a non-trivial nullspace $N(\lambda -T)$ which is called eigenspace of $T$ to the eigenvalue $\lambda$, i.e. $\lambda -T$ is not injective and hence cannot be inverted. The dimension of $N(\lambda -T)$ is called multiplicity of $\lambda$, and an element $\psi \in N(\lambda -T)-\{0\}$ is called an eigenelement or eigenvector of or to the eigenvalue $\lambda$.

In case $\lambda$ is no eigenvalue, then the operator $\lambda -T$ is injective and we can define
$$
R(\lambda,T)=(\lambda-T)^{-1}
$$
The set
$$
\varrho(T) = \{\,\lambda \in \mathbb{F}\, : \,\lambda-T \text{ is injective and } R (\lambda , T)\in \mathcal{B}(\mathcal{H})\,\}
$$
is called set of resolvents of $T$. If $T$ isn’t closed, then neither are $\lambda -T$ and $R(\lambda,T)$, which means that the set of resolvents is empty: $\varrho(T)=\emptyset$. Therefore we will assume $T$ to be closed in this section and thus
$$ \varrho(T) = \{\,\lambda \in \mathbb{F}\, : \,\lambda -T\text{ is bijective}\,\} $$

The function
\begin{equation}
\begin{aligned}
R(*,T)\, : \,\varrho(T) &\longrightarrow \mathcal{B}(\mathcal{H})\\
\lambda & \longmapsto R(\lambda,T)
\end{aligned}
\end{equation}
is called resolvent function of $T$ and the operator $R(\lambda,T)$ for a $\lambda \in \varrho(T)$ is called a resolvent of $T$ at $\lambda$. The complement
\begin{equation}
\sigma(T) = \mathbb{F}-\varrho(T) = \varrho(T)^{C_\mathbb{F}}
\end{equation}
is called the spectrum of the operator $T$. The spectrum contains the eigenvalues $\sigma_p(T)$ of $T$ and $\sigma_p(T)$ is called the point spectrum of $T$. So for the number $\lambda \in \sigma(T)$ the operator $\lambda -T$ is either not invertible or unbounded.

As we used overline for complex conjugates as well as for topological closures, let us now write the complex conjugate as $\iota : \lambda \mapsto \iota(\lambda)= \overline{\lambda}$ to avoid confusion as we want to apply it on sets, i.e. $\iota(A)=\{\iota(\alpha)= \overline{\alpha}\,\vert \,\alpha \in A \subseteq \mathbb{C} \}$.

Let $S,T\, : \,\mathcal{H}\longrightarrow \mathcal{H}$ be linear, closed operators on a Hilbert space, $\lambda \in \varrho(S)\cap \varrho(T)\, , \,\lambda_1,\lambda_2 \in \varrho(T)$. Then

\begin{equation*}
\sigma(T\,^*)=\iota(\sigma(T))\,\, \text{ and } \,\, \varrho(T\,^*)=\iota(\varrho(T)) \end{equation*}

First Resolvent Identity.

\begin{equation*} R(\lambda_1,T)-R(\lambda_2,T)= (\lambda_2-\lambda_1)R(\lambda_1,T)R(\lambda_2,T) \\ [R(\lambda_1,T),R(\lambda_2,T)] = R(\lambda_1,T)R(\lambda_2,T)- R(\lambda_2,T)R(\lambda_1,T)=0 \end{equation*}

Second Resolvent Identity.

\begin{equation*} D(S)\subseteq D(T)\,:\,R(\lambda,T)-R(\lambda,S)=R(\lambda,T)(T-S)R(\lambda,S) \\ D(S)=D(T)\,:\,R(\lambda,T)-R(\lambda,S)=R(\lambda,T)(T-S)R(\lambda,S) \\ =R(\lambda,S)(T-S)R(\lambda,T)  \end{equation*}

The eigenvalues of a Hermitian operator are real, and eigenelements to distinct eigenvalues of a Hermitian or normal operator are orthogonal.

 

Table I. Linearity And Topology.

\begin{array}{l,c,l} &&\\ \text{linear}&-& T(\alpha \psi + \beta \chi) = \alpha T\psi+\beta T\chi \\ &&\\ \text{finite dimensional }&-& \operatorname{dim} R(T) < \infty \\ &&\\ \text{continuous}&-&\lim_{n \to \infty}\psi_n = \psi \, \Longrightarrow \,\lim_{n \to \infty}T\psi_n = T\psi \\&&\\ \text{bounded}&-& ||T\psi||\leq C\cdot||\psi||\; , \;T\in \mathcal{B}(\mathcal{H})\\ &&\\ \text{continuation}&-& T\,: \,D(S)\subseteq D(T)\, , \, S\psi = T\psi \\ && \\ \text{restriction}&-& S\, : \,D(S)\subseteq D(T)\, , \,S\psi = T\psi\\ &&\\ \text{formally adjoint}&-&T\, : \,\mathcal{H}_1 \longrightarrow \mathcal{H}_2\,\,\text{ and }\,\, T\,^*\, : \,\mathcal{H}_2 \longrightarrow \mathcal{H}_1\\&& \langle \psi,T\chi \rangle_2 = \langle T\,^*\psi,\chi\rangle_1 \\ &&\\ \text{adjoint} & – & T\, : \,\mathcal{H}_1 \longrightarrow \mathcal{H}_2\,\,\text{ and }\,\, T\,^*\, : \,\mathcal{H}_2 \longrightarrow \mathcal{H}_1 \\&&\langle \psi,T\chi \rangle_2 = \langle T\,^*\psi,\chi\rangle_1 \text{ and }\, T\,^* \text{ is the maximal }\\&&\text{continuation of all formally adjoint operators.} \\ &&\\  \text{densely defined}&-&\ \overline{D(T)}=\mathcal{H}\\ &&\\ \text{Hermitian}&-&\langle T\psi,\chi \rangle = \langle \psi,T\chi \rangle \\&&\\ \text{symmetric}&-&\langle T\psi , \chi \rangle = \langle \psi , T \chi \rangle \text{ and }\overline{D(T)}=\mathcal{H}\\ &&\\ \text{self-adjoint}&-& \langle T\psi,\chi \rangle = \langle \psi,T\chi \rangle \text{ and }\overline{D(T)}=\mathcal{H} \text{ and } T\,^*=T \\ &&\\ \text{normal} & – & D(T\,^*)=D(T) \; , \;||T\psi||=||T\,^*\psi|| \; , \;\overline{D(T)}=\mathcal{H} \\&&\\ \end{array}

 

Table II. Spectrum.

\begin{array}{l,c, l}
&&\\
\text{closed}&-&G(T)=\{(\psi,T\psi)\} \text{ is closed}\\
&&\\
\text{closable}&-&\overline{G(T)}\text{ is the graph of a unique }  \overline{T} \\ &&\text{i.e. }G(\overline{T})=\overline{G(T)}\\
&&\\
\text{closure}&-&\overline{T}\\
&&\\ \text{core or essential domain}&-&D \subseteq D(T)\, : \,T=\overline{T}=\overline{T|_D}\\ &&\\ \text{T-norm or graph norm}&-&\langle \psi ,\chi \rangle_T = \langle \psi,\chi \rangle + \langle T\psi, T\chi \rangle \\ &&||\psi||_T=\left(||\psi||^2+||T\psi||^2\right)^{1/2}\\ &&\\ \text{compact}&-&\text{For all bounded sequences in }D(T) \\ &&\text{there is a subsequence }(\psi_n)_n\subseteq D(T) \\
&&\text{such that } \operatorname{lim}_{n \to \infty}T\psi_n \text{ exists.}\\ &&\\ \text{eigenvalue}&-&T\psi = \lambda \cdot \psi \\ &&\\ \text{eigen space}&-&N(\lambda – T)\\ &&\\  \text{multiplicity}&-&\operatorname{dim}N(\lambda -T)\\ &&\\ \text{eigen vector or element}&-&\psi \in N(\lambda -T)-\{0\} \\ && \\ \text{resolvent set}&-&\varrho(T) = \{\lambda \in  \mathbb{F} : \lambda-T \text{ is injective and …} \\ &&\quad \quad\quad \ldots \, R(\lambda , T)=(\lambda -T)^{-1}\in \mathcal{B}(\mathcal{H})\,\}\\ & &\\ \text{resolvent function}&-& R(*,T)\, : \,\varrho(T)\longrightarrow \mathcal{B}(\mathcal{H}) \\ &&\\ \text{resolvent} &-& R(\lambda,T)=(\lambda-T)^{-1}\text{ bounded}\\ &&\\ \text{spectrum}&-& \sigma(T)=\mathbb{F}-\varrho(T)\\ &&\\ \text{point spectrum}&-& \text{eigenvalues } \sigma_p(T) \subseteq \sigma(T)\text{ of }T \end{array}

 

 

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