Spectrum (functional analysis)

 ( Spectral Theory ) 

In mathematics, particularly in functional analysis, the spectrum of a bounded operator (or, more generally, an unbounded linear operator) is a generalisation of the set of of a matrix. Specifically, a complex number $\backslash lambda$ is said to be in the spectrum of a bounded linear operator $T$ if $T-\backslash lambda\; I$

• either has no set-theoretic inverse function;
• or the set-theoretic inverse is either unbounded or defined on a non-dense subset.

Here, $I$ is the identity operator.

By the closed graph theorem, $\backslash lambda$ is in the spectrum if and only if the bounded operator $T\; -\; \backslash lambda\; I:\; V\backslash to\; V$ is non-bijective on $V$.

The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.

The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the unilateral shift operator R on the Hilbert space Lp space,

$(x\_1,\; x\_2,\; \backslash dots)\; \backslash mapsto\; (0,\; x\_1,\; x\_2,\; \backslash dots).$

This has no eigenvalues, since if Rx= λx then by expanding this expression we see that x₁=0, x₂=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in Lp space. In fact every bounded linear operator on a complex number Banach space must have a non-empty spectrum.

The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator $T:\backslash ,X\backslash to\; X$ defined on domain $D(T)\backslash subseteq\; X$ if there is no bounded inverse $(T-\backslash lambda\; I)^\{-1\}:\backslash ,X\backslash to\; D(T)$ defined on the whole of $X.$ If T is closed operator (which includes the case when T is bounded), boundedness of $(T-\backslash lambda\; I)^\{-1\}$ follows automatically from its existence.

The space of bounded linear operators B( X) on a Banach space X is an example of a unital algebra Banach algebra. Since the definition of the spectrum does not mention any properties of B( X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

---

Spectrum of a bounded operator

---

Definition Let $T$ be a bounded linear operator acting on a Banach space $X$ over the complex scalar field $\backslash mathbb\{C\}$, and $I$ be the identity operator on $X$. The spectrum of $T$ is the set of all $\backslash lambda\; \backslash in\; \backslash mathbb\{C\}$ for which the operator $T-\backslash lambda\; I$ does not have an inverse that is a bounded linear operator.

Since $T-\backslash lambda\; I$ is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem, it is bounded. Therefore, the spectrum consists precisely of those scalars $\backslash lambda$ for which $T-\backslash lambda\; I$ is not bijective.

The spectrum of a given operator $T$ is often denoted $\backslash sigma(T)$, and its complement, the resolvent set, is denoted $\backslash rho(T)\; =\; \backslash mathbb\{C\}\; \backslash setminus\; \backslash sigma(T)$. ($\backslash rho(T)$ is sometimes used to denote the spectral radius of $T$)

---

Relation to eigenvalues If $\backslash lambda$ is an eigenvalue of $T$, then the operator $T-\backslash lambda\; I$ is not one-to-one, and therefore its inverse $(T-\backslash lambda\; I)^\{-1\}$ is not defined. However, the converse statement is not true: the operator $T\; -\; \backslash lambda\; I$ may not have an inverse, even if $\backslash lambda$ is not an eigenvalue. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.

For example, consider the Hilbert space $\backslash ell^2(\backslash Z)$, that consists of all bi-infinite sequences of real numbers

$v\; =\; (\backslash ldots,\; v\_\{-2\},v\_\{-1\},v\_0,v\_1,v\_2,\backslash ldots)$

that have a finite sum of squares $\backslash sum\_\{i=-\backslash infty\}^\{+\backslash infty\}\; v\_i^2$. The bilateral shift operator $T$ simply displaces every element of the sequence by one position; namely if $u\; =\; T(v)$ then $u\_i\; =\; v\_\{i-1\}$ for every integer $i$. The eigenvalue equation $T(v)\; =\; \backslash lambda\; v$ has no nonzero solution in this space, since it implies that all the values $v\_i$ have the same absolute value (if $\backslash vert\; \backslash lambda\; \backslash vert\; =\; 1$) or are a geometric progression (if $\backslash vert\; \backslash lambda\; \backslash vert\; \backslash neq\; 1$); either way, the sum of their squares would not be finite. However, the operator $T-\backslash lambda\; I$ is not invertible if $|\backslash lambda|\; =\; 1$. For example, the sequence $u$ such that $u\_i\; =\; 1/(|i|+1)$ is in $\backslash ell^2(\backslash Z)$; but there is no sequence $v$ in $\backslash ell^2(\backslash Z)$ such that $(T-I)v\; =\; u$ (that is, $v\_\{i-1\}\; =\; u\_i\; +\; v\_i$ for all $i$).

---

Basic properties The spectrum of a bounded operator T is always a closed set, bounded set subset of the complex plane.

If the spectrum were empty, then the resolvent function

$R(\backslash lambda)\; =\; (T-\backslash lambda\; I)^\{-1\},\; \backslash qquad\; \backslash lambda\backslash in\backslash Complex,$

would be defined everywhere on the complex plane and bounded. But it can be shown that the resolvent function R is holomorphic on its domain. By the vector-valued version of Liouville's theorem, this function is constant, thus everywhere zero as it is zero at infinity. This would be a contradiction.

The boundedness of the spectrum follows from the Neumann series in λ; the spectrum σ( T) is bounded by || T||. A similar result shows the closedness of the spectrum.

The bound || T|| on the spectrum can be refined somewhat. The spectral radius, r( T), of T is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum σ( T) inside of it, i.e.

$r(T)\; =\; \backslash sup\; \backslash$>

The spectral radius formula saysTheorem 3.3.3 of Kadison & Ringrose, 1983, Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory, New York: Academic Press, Inc. that for any element $T$ of a Banach algebra, $r(T)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left\backslash |T^n\backslash right\backslash ^\{1/n\}.$ Spectrum of an unbounded operator One can extend the definition of spectrum to unbounded operators on a Banach space X. These operators are no longer elements in the Banach algebra B( X). Definition Let X be a Banach space and $T:\backslash ,D(T)\backslash to\; X$ be a linear operator defined on domain $D(T)\; \backslash subseteq\; X$. A complex number λ is said to be in the resolvent set (also called regular set) of $T$ if the operator $T-\backslash lambda\; I:\backslash ,D(T)\; \backslash to\; X$ has a bounded everywhere-defined inverse, i.e. if there exists a bounded operator $S\; :\backslash ,\; X\; \backslash rightarrow\; D(T)$ such that $S\; (T\; -\; \backslash lambda\; I)\; =\; I\_\{D(T)\},\; \backslash ,\; (T\; -\; \backslash lambda\; I)\; S\; =\; I\_X.$ A complex number λ is then in the spectrum if λ is not in the resolvent set. For λ to be in the resolvent (i.e. not in the spectrum), just like in the bounded case, $T-\backslash lambda\; I$ must be bijective, since it must have a two-sided inverse. As before, if an inverse exists, then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately. By the closed graph theorem, boundedness of $(T-\backslash lambda\; I)^\{-1\}$ does follow directly from its existence when T is closed operator. Then, just as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if $T-\backslash lambda\; I$ is not bijective. Note that the class of closed operators includes all bounded operators. Basic properties The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane. If the operator T is not closed, then $\backslash sigma(T)=\backslash Complex$. The following example indicates that non-closed operators may have empty spectra. Let $T$ denote the differentiation operator on $L^2(0,1)$, whose domain is defined to be the closure of $C^\{\backslash infty\}\_c((0,1])$ with respect to the $H^1$-Sobolev space norm. This space can be characterized as all functions in $H^1(0,1)$ that are zero at $t\; =\; 0$. Then, $T\; -\; z$ has trivial kernel on this domain, as any $H^1(0,1)$-function in its kernel is a constant multiple of $e^\{zt\}$, which is zero at $t\; =\; 0$ if and only if it is identically zero. Therefore, the complement of the spectrum is all of $\backslash mathbb\{C\}.$ Classification of points in the spectrum A bounded operator T on a Banach space is invertible, i.e. has a bounded inverse, if and only if T is bounded below, i.e. $\backslash$

, for some $c\; >\; 0,$ and has dense range. Accordingly, the spectrum of T can be divided into the following parts: $\backslash lambda\backslash in\backslash sigma(T)$ if $T\; -\; \backslash lambda\; I$ is not bounded below. In particular, this is the case if $T\; -\; \backslash lambda\; I$ is not injective, that is, λ is an eigenvalue. The set of eigenvalues is called the point spectrum of T and denoted by σp( T). Alternatively, $T-\backslash lambda\; I$ could be one-to-one but still not bounded below. Such λ is not an eigenvalue but still an approximate eigenvalue of T (eigenvalues themselves are also approximate eigenvalues). The set of approximate eigenvalues (which includes the point spectrum) is called the approximate point spectrum of T, denoted by σap( T). $\backslash lambda\backslash in\backslash sigma(T)$ if $T-\backslash lambda\; I$ does not have dense range. The set of such λ is called the compression spectrum of T, denoted by $\backslash sigma\_\{\backslash mathrm\{cp\}\}(T)$. If $T-\backslash lambda\; I$ does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by $\backslash sigma\_\{\backslash mathrm\{r\}\}(T)$. Note that the approximate point spectrum and residual spectrum are not necessarily disjoint (however, the point spectrum and the residual spectrum are). The following subsections provide more details on the three parts of σ( T) sketched above. Point spectrum If an operator is not injective (so there is some nonzero x with T( x) = 0), then it is clearly not invertible. So if λ is an eigenvalue of T, one necessarily has λ ∈  σ( T). The set of eigenvalues of T is also called the point spectrum of T, denoted by σp( T). Some authors refer to the closure of the point spectrum as the pure point spectrum $\backslash sigma\_\{pp\}(T)=\backslash overline\{\backslash sigma\_\{p\}(T)\}$ while others simply consider $\backslash sigma\_\{pp\}(T):=\backslash sigma\_\{p\}(T).$ Approximate point spectrum More generally, by the bounded inverse theorem, T is not invertible if it is not bounded below; that is, if there is no c > 0 such that

for all . So the spectrum includes the set of approximate eigenvalues, which are those λ such that is not bounded below; equivalently, it is the set of λ for which there is a sequence of unit vectors x1, x2, ... for which $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash$

= 0. The set of approximate eigenvalues is known as the approximate point spectrum, denoted by $\backslash sigma\_\{\backslash mathrm\{ap\}\}(T)$. It is easy to see that the eigenvalues lie in the approximate point spectrum. For example, consider the bilateral shift W on $l^2(\backslash Z)$ defined by $W:\backslash ,e\_j\backslash mapsto\; e\_\{j+1\},\backslash quad\; j\backslash in\backslash Z,$ where $\backslash big(e\_j\backslash big)\_\{j\backslash in\backslash N\}$ is the standard orthonormal basis in $l^2(\backslash Z)$. Direct calculation shows W has no eigenvalues, but every λ with $$

=1 is an approximate eigenvalue; letting x n be the vector $\backslash frac\{1\}\{\backslash sqrt\{n\}\}(\backslash dots,\; 0,\; 1,\; \backslash lambda^\{-1\},\; \backslash lambda^\{-2\},\; \backslash dots,\; \backslash lambda^\{1\; -\; n\},\; 0,\; \backslash dots)$ one can see that

= 1 for all n, but $\backslash$

= \sqrt{\frac{2}{n}} \to 0. Since W is a unitary operator, its spectrum lies on the unit circle. Therefore, the approximate point spectrum of W is its entire spectrum. This conclusion is also true for a more general class of operators. A unitary operator is normal operator. By the spectral theorem, a bounded operator on a Hilbert space H is normal if and only if it is equivalent (after identification of H with an $L^2$ space) to a multiplication operator. It can be shown that the approximate point spectrum of a bounded multiplication operator equals its spectrum. Discrete spectrum The discrete spectrum is defined as the set of normal eigenvalues or, equivalently, as the set of isolated points of the spectrum such that the corresponding Riesz projector is of finite rank. As such, the discrete spectrum is a strict subset of the point spectrum, i.e., $\backslash sigma\_d(T)\; \backslash subset\; \backslash sigma\_p(T).$ Continuous spectrum The set of all λ for which $T-\backslash lambda\; I$ is injective and has dense range, but is not surjective, is called the continuous spectrum of T, denoted by $\backslash sigma\_\{\backslash mathbb\{c\}\}(T)$. The continuous spectrum therefore consists of those approximate eigenvalues which are not eigenvalues and do not lie in the residual spectrum. That is, $\backslash sigma\_\{\backslash mathrm\{c\}\}(T)\; =\; \backslash sigma\_\{\backslash mathrm\{ap\}\}(T)\; \backslash setminus\; (\backslash sigma\_\{\backslash mathrm\{r\}\}(T)\; \backslash cup\; \backslash sigma\_\{\backslash mathrm\{p\}\}(T))$. For example, $A:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $e\_j\backslash mapsto\; e\_j/j$, $j\backslash in\backslash N$, is injective and has a dense range, yet $\backslash mathrm\{Ran\}(A)\backslash subsetneq\; l^2(\backslash N)$. Indeed, if $x\; =\; \backslash sum\_\{j\backslash in\backslash N\}\; c\_j\; e\_j\backslash in\; l^2(\backslash N)$ with $c\_j\; \backslash in\; \backslash Complex$ such that $\backslash sum\_\{j\backslash in\backslash N\}$

^2 < \infty, and then $\backslash sum\_\{j\backslash in\backslash N\}\; j\; c\_j\; e\_j\; \backslash notin\; l^2(\backslash N)$. Compression spectrum The set of $\backslash lambda\backslash in\backslash Complex$ for which $T-\backslash lambda\; I$ does not have dense range is known as the compression spectrum of T and is denoted by $\backslash sigma\_\{\backslash mathrm\{cp\}\}(T)$. Residual spectrum The set of $\backslash lambda\backslash in\backslash Complex$ for which $T-\backslash lambda\; I$ is injective but does not have dense range is known as the residual spectrum of T and is denoted by $\backslash sigma\_\{\backslash mathrm\{r\}\}(T)$: $\backslash sigma\_\{\backslash mathrm\{r\}\}(T)\; =\; \backslash sigma\_\{\backslash mathrm\{cp\}\}(T)\; \backslash setminus\; \backslash sigma\_\{\backslash mathrm\{p\}\}(T).$ An operator may be injective, even bounded below, but still not invertible. The right shift on $l^2(\backslash mathbb\{N\})$, $R:\backslash ,l^2(\backslash mathbb\{N\})\backslash to\; l^2(\backslash mathbb\{N\})$, $R:\backslash ,e\_j\backslash mapsto\; e\_\{j+1\},\backslash ,j\backslash in\backslash N$, is such an example. This shift operator is an isometry, therefore bounded below by 1. But it is not invertible as it is not surjective ($e\_1\backslash not\backslash in\backslash mathrm\{Ran\}(R)$), and moreover $\backslash mathrm\{Ran\}(R)$ is not dense in $l^2(\backslash mathbb\{N\})$ ($e\_1\backslash notin\backslash overline\{\backslash mathrm\{Ran\}(R)\}$). Peripheral spectrum The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius. (2025). 9783642606373, Springer Science & Business Media. . ISBN 9783642606373 Essential spectrum There are five similar definitions of the essential spectrum of closed densely defined linear operator $A\; :\; \backslash ,X\; \backslash to\; X$ which satisfy $$ \sigma_{\mathrm{ess},1}(A) \subset \sigma_{\mathrm{ess},2}(A) \subset \sigma_{\mathrm{ess},3}(A) \subset \sigma_{\mathrm{ess},4}(A) \subset \sigma_{\mathrm{ess},5}(A) \subset \sigma(A). All these spectra $\backslash sigma\_\{\backslash mathrm\{ess\},k\}(A),\backslash \; 1\backslash le\; k\backslash le\; 5$, coincide in the case of self-adjoint operators. The essential spectrum $\backslash sigma\_\{\backslash mathrm\{ess\},1\}(A)$ is defined as the set of points $\backslash lambda$ of the spectrum such that $A-\backslash lambda\; I$ is not semi-Fredholm. (The operator is semi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) Example 1: $\backslash lambda=0\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},1\}(A)$ for the operator $A:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $A:\backslash ,e\_j\backslash mapsto\; e\_j/j,~\; j\backslash in\backslash N$ (because the range of this operator is not closed: the range does not include all of $l^2(\backslash N)$ although its closure does). Example 2: $\backslash lambda=0\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},1\}(N)$ for $N:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $N:\backslash ,v\backslash mapsto\; 0$ for any $v\backslash in\; l^2(\backslash N)$ (because both kernel and cokernel of this operator are infinite-dimensional). The essential spectrum $\backslash sigma\_\{\backslash mathrm\{ess\},2\}(A)$ is defined as the set of points $\backslash lambda$ of the spectrum such that the operator either $A-\backslash lambda\; I$ has infinite-dimensional kernel or has a range which is not closed. It can also be characterized in terms of Weyl's criterion: there exists a sequence $(x\_j)\_\{j\backslash in\backslash N\}$ in the space X such that $\backslash Vert\; x\_j\backslash Vert=1$, $\backslash lim\_\{j\backslash to\backslash infty\}\; \backslash left\backslash$

= 0, and such that $(x\_j)\_\{j\backslash in\backslash N\}$ contains no convergent subsequence. Such a sequence is called a singular sequence (or a singular Weyl sequence). Example: $\backslash lambda=0\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},2\}(B)$ for the operator $B:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $B:\backslash ,e\_j\backslash mapsto\; e\_\{j/2\}$ if j is even and $e\_j\backslash mapsto\; 0$ when j is odd (kernel is infinite-dimensional; cokernel is zero-dimensional). Note that $\backslash lambda=0\backslash not\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},1\}(B)$. The essential spectrum $\backslash sigma\_\{\backslash mathrm\{ess\},3\}(A)$ is defined as the set of points $\backslash lambda$ of the spectrum such that $A-\backslash lambda\; I$ is not Fredholm. (The operator is Fredholm if its range is closed and both its kernel and cokernel are finite-dimensional.) Example: $\backslash lambda=0\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},3\}(J)$ for the operator $J:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $J:\backslash ,e\_j\backslash mapsto\; e\_\{2j\}$ (kernel is zero-dimensional, cokernel is infinite-dimensional). Note that $\backslash lambda=0\backslash not\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},2\}(J)$. The essential spectrum $\backslash sigma\_\{\backslash mathrm\{ess\},4\}(A)$ is defined as the set of points $\backslash lambda$ of the spectrum such that $A-\backslash lambda\; I$ is not Fredholm of index zero. It could also be characterized as the largest part of the spectrum of A which is preserved by compact operator perturbations. In other words, $\backslash sigma\_\{\backslash mathrm\{ess\},4\}(A)\; =\; \backslash bigcap\_\{K\; \backslash in\; B\_0(X)\}\; \backslash sigma(A+K)$; here $B\_0(X)$ denotes the set of all compact operators on X. Example: $\backslash lambda=0\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},4\}(R)$ where $R:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$ is the right shift operator, $R:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $R:\backslash ,e\_j\backslash mapsto\; e\_\{j+1\}$ for $j\backslash in\backslash N$ (its kernel is zero, its cokernel is one-dimensional). Note that $\backslash lambda=0\backslash not\backslash in\backslash sigma\_\{\backslash mathrm\{ess\},3\}(R)$. The essential spectrum $\backslash sigma\_\{\backslash mathrm\{ess\},5\}(A)$ is the union of $\backslash sigma\_\{\backslash mathrm\{ess\},1\}(A)$ with all components of $\backslash Complex\; \backslash setminus\; \backslash sigma\_\{\backslash mathrm\{ess\},1\}(A)$ that do not intersect with the resolvent set $\backslash Complex\; \backslash setminus\; \backslash sigma(A)$. It can also be characterized as $\backslash sigma(A)\backslash setminus\backslash sigma\_\{\backslash mathrm\{d\}\}(A)$. Example: consider the operator $T:\backslash ,l^2(\backslash Z)\backslash to\; l^2(\backslash Z)$, $T:\backslash ,e\_j\backslash mapsto\; e\_\{j-1\}$ for $j\backslash ne\; 0$, $T:\backslash ,e\_0\backslash mapsto\; 0$. Since $\backslash Vert\; T\backslash Vert=1$, one has $\backslash sigma(T)\backslash subset\backslash overline\{\backslash mathbb\{D\}\_1\}$. For any $z\backslash in\backslash Complex$ with $$

\le 1\}. Example: Hydrogen atom The hydrogen atom provides an example of different types of the spectra. The hydrogen atom Hamiltonian operator $H=-\backslash Delta-\backslash frac\{Z\}\{$

, $Z\; >\; 0$, with domain $D(H)\; =\; H^1(\backslash R^3)$ has a discrete set of eigenvalues (the discrete spectrum $\backslash sigma\_\{\backslash mathrm\{d\}\}(H)$, which in this case coincides with the point spectrum $\backslash sigma\_\{\backslash mathrm\{p\}\}(H)$ since there are no eigenvalues embedded into the continuous spectrum) that can be computed by the Rydberg formula. Their corresponding are called eigenstates, or the . The result of the ionization process is described by the continuous part of the spectrum (the energy of the collision/ionization is not "quantized"), represented by $\backslash sigma\_\{\backslash mathrm\{cont\}\}(H)=[0,+\backslash infty)$ (it also coincides with the essential spectrum, $\backslash sigma\_\{\backslash mathrm\{ess\}\}(H)=[0,+\backslash infty)$).

---

Spectrum of the adjoint operator

Let X be a Banach space and $T:\backslash ,X\backslash to\; X$ a closed linear operator with dense domain $D(T)\backslash subset\; X$. If X* is the dual space of X, and $T^*:\backslash ,\; X^*\; \backslash to\; X^*$ is the hermitian adjoint of T, then

$\backslash sigma(T^*)\; =\; \backslash overline\{\backslash sigma(T)\}\; :=\; \backslash \{z\backslash in\backslash Complex\; :\; \backslash bar\{z\}\backslash in\backslash sigma(T)\backslash \}.$

(T) = \overline{\sigma_{\mathrm{p}}(T^*)}.

We also get $\backslash sigma\_\{\backslash mathrm\{p\}\}(T)\backslash subset\backslash overline\{\backslash sigma\_\{\backslash mathrm\{r\}\}(T^*)\backslash cup\; \backslash sigma\_\{\backslash mathrm\{p\}\}(T^*)\}$ by the following argument: X embeds isometrically into X**. Therefore, for every non-zero element in the kernel of $T-\backslash lambda\; I$ there exists a non-zero element in X** which vanishes on $\backslash mathrm\{Ran\}(T^*\; -\; \backslash bar\{\backslash lambda\}I)$. Thus $\backslash mathrm\{Ran\}(T^*\; -\backslash bar\{\backslash lambda\}\; I)$ can not be dense.

Furthermore, if X is reflexive, we have $\backslash overline\{\backslash sigma\_\{\backslash mathrm\{r\}\}(T^*)\}\backslash subset\backslash sigma\_\{\backslash mathrm\{p\}\}(T)$.

---

Spectra of particular classes of operators

---

Compact operators

If T is a compact operator, or, more generally, an inessential operator, then it can be shown that the spectrum is countable, that zero is the only possible accumulation point, and that any nonzero λ in the spectrum is an eigenvalue.

---

Quasinilpotent operators

A bounded operator $A:\backslash ,X\backslash to\; X$ is quasinilpotent if $\backslash lVert\; A^n\backslash rVert^\{1/n\}\; \backslash to\; 0$ as $n\backslash to\backslash infty$ (in other words, if the spectral radius of A equals zero). Such operators could equivalently be characterized by the condition

$\backslash sigma(A)=\backslash \{0\backslash \}.$

An example of such an operator is $A:\backslash ,l^2(\backslash N)\backslash to\; l^2(\backslash N)$, $e\_j\backslash mapsto\; e\_\{j+1\}/2^j$ for $j\backslash in\backslash N$.

---

Self-adjoint operators

If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).

For self-adjoint operators, one can use to define a decomposition of the spectrum into absolutely continuous, pure point, and singular parts.

---

Spectrum of a real operator

The definitions of the resolvent and spectrum can be extended to any continuous linear operator $T$ acting on a Banach space $X$ over the real field $\backslash mathbb\{R\}$ (instead of the complex field $\backslash mathbb\{C\}$) via its complexification $T\_\backslash mathbb\{C\}$. In this case we define the resolvent set $\backslash rho(T)$ as the set of all $\backslash lambda\backslash in\backslash mathbb\{C\}$ such that $T\_\backslash mathbb\{C\}-\backslash lambda\; I$ is invertible as an operator acting on the complexified space $X\_\backslash mathbb\{C\}$; then we define $\backslash sigma(T)=\backslash mathbb\{C\}\backslash setminus\backslash rho(T)$.

---

Real spectrum

The real spectrum of a continuous linear operator $T$ acting on a real Banach space $X$, denoted $\backslash sigma\_\backslash mathbb\{R\}(T)$, is defined as the set of all $\backslash lambda\backslash in\backslash mathbb\{R\}$ for which $T-\backslash lambda\; I$ fails to be invertible in the real algebra of bounded linear operators acting on $X$. In this case we have $\backslash sigma(T)\backslash cap\backslash mathbb\{R\}=\backslash sigma\_\backslash mathbb\{R\}(T)$. Note that the real spectrum may or may not coincide with the complex spectrum. In particular, the real spectrum could be empty.

---

Spectrum of a unital Banach algebra

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ( x) (or more explicitly σ _B( x)) of an element x of B to be the set of those λ for which λe −  x is not invertible in B. This extends the definition for bounded linear operators B( X) on a Banach space X, since B( X) is a unital Banach algebra.

---

See also

• Essential spectrum
• Discrete spectrum (mathematics)
• Self-adjoint operator
• Pseudospectrum
• Resolvent set

---

Notes

• Dales et al., Introduction to Banach Algebras, Operators, and Harmonic Analysis,
• (2025). 9780821834466, American Mathematical Society. ISBN 9780821834466
• G. Teschl (2025). 9781470417048, American Mathematical Soc.. ISBN 9781470417048

Categories: Spectral Theory

Post Comment