Fact-checked by Grok 2 weeks ago

Resolvent set

In functional analysis, the resolvent set of a bounded linear operator A on a complex Banach space X is the set \rho(A) = \{\lambda \in \mathbb{C} \mid \lambda I - A is bijective from X onto X with bounded inverse

\}\), where

Idenotes the [identity](/page/Identity) [operator](/page/Operator) onX.[1] The corresponding **resolvent operator** is defined as R(\lambda, A) = (\lambda I - A)^{-1}, which belongs to the space B(X)of bounded linear [operator](/page/Operator)s onXfor each\lambda \in \rho(A)$.^[1] The resolvent set \rho(A) is the complement in \mathbb{C} of the spectrum \sigma(A) of A, and \sigma(A) is a nonempty compact subset of the closed disk of radius \|A\| centered at the origin.^[1] For bounded operators, \rho(A) is always open in \mathbb{C}, and the resolvent R(\cdot, A): \rho(A) \to B(X) is a holomorphic function, satisfying the resolvent identity R(\lambda, A) - R(\mu, A) = (\mu - \lambda) R(\lambda, A) R(\mu, A) for distinct \lambda, \mu \in \rho(A).^[1] This holomorphy enables the use of complex analysis techniques, such as contour integrals, to study spectral projections and decompositions of the operator.^[2] For unbounded linear operators, the notion extends to densely defined closed operators A: D(A) \subseteq X \to X, where \rho(A) consists of those \lambda \in \mathbb{C} such that \lambda I - A: D(A) \to X is bijective with a bounded inverse defined on all of X.^[2] In this case, the resolvent set remains open, and the resolvent operator inherits similar analytic properties, playing a central role in the spectral theory of differential operators and evolution equations.^[2] The resolvent norm \|R(\lambda, A)\| provides insights into the pseudospectrum and stability of the operator, with level sets \{\lambda \in \mathbb{C} \mid \|R(\lambda, A)\| = 1/\epsilon\} influencing applications in numerical analysis and perturbation theory.^[2]

Background Concepts

Linear Operators on Banach Spaces

A Banach space is defined as a normed vector space over the complex numbers that is complete with respect to the metric induced by its norm, meaning every Cauchy sequence converges to an element within the space.^[3] This completeness property ensures that limits of convergent sequences remain in the space, providing a robust framework for analysis similar to the real numbers but in infinite dimensions.^[4] Common examples include the sequence spaces \ell^p for $1 \leq p \leq \infty, where the norm is \|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p} (with the supremum norm for p=\infty), and the function spaces L^p(\mu) consisting of measurable functions with finite \int |f|^p \, d\mu < \infty.^[5] A linear operator T: X \to X on a Banach space X is a map that preserves vector addition and scalar multiplication, i.e., T(\alpha x + y) = \alpha T x + T y for all scalars \alpha \in \mathbb{C} and x, y \in X. Such an operator is bounded if there exists a finite constant M such that \|T x\| \leq M \|x\| for all x \in X, and its operator norm is given by

\|T\| = \sup_{\|x\| \leq 1} \|T x\| = \sup_{x \neq 0} \frac{\|T x\|}{\|x\|} < \infty.

This norm measures the maximum "stretching" effect of T, and boundedness is equivalent to continuity of T with respect to the norm topology.^[4]^[6] Representative examples of bounded linear operators include multiplication operators on L^p spaces. For a measurable function m \in L^\infty(\mu) with \|m\|_\infty = \esssup |m| < \infty, the operator M_m: L^p(\mu) \to L^p(\mu) defined by (M_m f)(x) = m(x) f(x) satisfies \|M_m\| = \|m\|_\infty, making it bounded. Another class is integral operators, such as the Volterra operator on C([0,1]) or L^p([0,1]), given by

(K f)(x) = \int_0^x k(x,t) f(t) \, dt,

where the kernel k is continuous; under suitable conditions on k, K is compact and thus bounded.^[4] The collection of all bounded linear operators on a Banach space X, denoted B(X), forms a Banach algebra under the operations of composition (multiplication) and addition, equipped with the operator norm. Specifically, B(X) is complete as a normed space, and the norm satisfies \|S T\| \leq \|S\| \|T\| for S, T \in B(X), with the identity operator serving as the multiplicative unit.^[7] This algebraic structure underpins much of operator theory, enabling the study of invertibility and other properties through algebraic and analytic tools.^[4]

The Spectrum of an Operator

In the context of bounded linear operators on Banach spaces, the resolvent set of an operator T, denoted \rho(T), consists of all complex numbers \lambda \in \mathbb{C} such that the operator \lambda I - T is bijective and its inverse is bounded.^[8] This condition ensures that \lambda I - T is invertible in the algebra of bounded operators, providing a framework for analyzing the behavior of T away from problematic values. The resolvent set plays a crucial role in operator theory by identifying points where T behaves "regularly," allowing for the application of inverse operator techniques in solving equations involving T. The spectrum of T, denoted \sigma(T), is defined as the complement \sigma(T) = \mathbb{C} \setminus \rho(T), comprising those \lambda \in \mathbb{C} for which \lambda I - T fails to be invertible, either because it is not injective, not surjective, or both.^[8] This failure of invertibility manifests in different ways, leading to a classification of the spectrum into three disjoint subsets: the point spectrum \sigma_p(T), the continuous spectrum \sigma_c(T), and the residual spectrum \sigma_r(T). The point spectrum consists of eigenvalues, where \lambda I - T is not injective, meaning there exists a non-zero vector in the kernel. The continuous spectrum includes points where \lambda I - T is injective but its range is dense yet not the entire space, while the residual spectrum covers cases where \lambda I - T is injective but the range is not dense.^[9] These categories provide a refined understanding of the singularities in the operator's action, with the full spectrum \sigma(T) = \sigma_p(T) \cup \sigma_c(T) \cup \sigma_r(T). A key quantitative measure associated with the spectrum is the spectral radius r(T) = \sup \{ |\lambda| : \lambda \in \sigma(T) \}, which Gelfand's formula equates to r(T) = \lim_{n \to \infty} \|T^n\|^{1/n}. This formula links the growth of powers of T to the size of its spectrum, offering a practical way to compute or bound the spectral radius without directly determining \sigma(T). For bounded operators T on infinite-dimensional complex Banach spaces, the spectrum \sigma(T) is always a non-empty compact subset of \mathbb{C}.^[10] To see non-emptiness, suppose for contradiction that \rho(T) = \mathbb{C}; then the resolvent function, when composed with bounded linear functionals, yields entire functions on \mathbb{C} that are bounded (since \|(\lambda I - T)^{-1}\| \leq 1/ \mathrm{dist}(\lambda, \sigma(T)) but here with empty spectrum implying uniform boundedness for large |\lambda|), and vanish at infinity, hence constant by Liouville's theorem, leading to a contradiction unless the space is trivial. Compactness follows from closedness (as the complement of the open resolvent set) and boundedness (enclosed in the disk of radius r(T) < \infty).

Definition

Formal Definition

In the context of a complex Banach space X and a bounded linear operator T: X \to X, the resolvent set of T, denoted \rho(T), is the subset of the complex plane \mathbb{C} consisting of all scalars \lambda \in \mathbb{C} such that the operator \lambda I - T: X \to X is bijective and possesses a bounded inverse.^[11] Bijectivity requires \lambda I - T to be both injective (i.e., \ker(\lambda I - T) = \{0\}) and surjective (i.e., \operatorname{ran}(\lambda I - T) = X); for bounded operators on Banach spaces, the open mapping theorem guarantees that any such bijective operator has a bounded inverse, ensuring the resolvent exists in the space of bounded operators B(X).[](https://loss.math.gatech.edu/14SPRINGTEA/spectral theory.pdf) For \lambda \in \rho(T), the bounded inverse is denoted by the resolvent operator R(\lambda, T) = (\lambda I - T)^{-1} \in B(X).^[11] The resolvent set \rho(T) is an open subset of \mathbb{C}. This follows from perturbation arguments: if \lambda_0 \in \rho(T), then for all \lambda \in \mathbb{C} satisfying |\lambda - \lambda_0| < 1 / \|R(\lambda_0, T)\|, the operator \lambda I - T is invertible via the Neumann series expansion R(\lambda, T) = R(\lambda_0, T) \sum_{n=0}^\infty [(\lambda_0 - \lambda) R(\lambda_0, T)]^n, which converges in the operator norm.^[12]

Resolvent Operator

The resolvent operator associated with a closed linear operator T on a Banach space X is defined, for each \lambda \in \rho(T), as the bounded linear operator R(\lambda, T) = (\lambda I - T)^{-1}: X \to X, which serves as the inverse of the shifted operator \lambda I - T.^[12] This operator maps the entire space X onto itself bijectively and continuously, ensuring that \lambda I - T is invertible in the sense of bounded operators on X.^[12] A fundamental bound on the norm of the resolvent operator is given by \|R(\lambda, T)\| \geq 1 / \dist(\lambda, \sigma(T)), where \sigma(T) denotes the spectrum of T and \dist(\lambda, \sigma(T)) = \inf_{\mu \in \sigma(T)} |\lambda - \mu|. This estimate reflects the resolvent's sensitivity to the proximity of \lambda to the spectrum, providing a quantitative measure of stability away from spectral points. Additionally, the resolvent satisfies the algebraic identity known as the resolvent equation: for \mu, \lambda \in \rho(T),

R(\lambda, T) - R(\mu, T) = (\mu - \lambda) R(\lambda, T) R(\mu, T).

^[12] This relation, derived from the invertibility of both \lambda I - T and \mu I - T, underscores the interconnectedness of resolvents at different points in the resolvent set. In the context of spectral theory, the resolvent operator R(\lambda, T) forms the basis for constructing the functional calculus, particularly through its role in generating resolvents from spectral measures in the spectral theorem for self-adjoint or normal operators. Specifically, for an operator admitting a spectral resolution T = \int_{\sigma(T)} \mu \, dE(\mu) with respect to a spectral measure E, the resolvent is expressed as R(\lambda, T) = \int_{\sigma(T)} (\lambda - \mu)^{-1} \, dE(\mu), linking the operator-valued function to the underlying spectral decomposition. As \lambda approaches the boundary \partial \rho(T) of the resolvent set, the norm \|R(\lambda, T)\| tends to infinity, owing to the diminishing distance to the spectrum and the reciprocal bound on the norm.^[12] This blow-up behavior highlights the resolvent's role in delineating the boundary between invertible and non-invertible regimes for the shifted operator.

Properties

Basic Properties

The resolvent set \rho(T) of a bounded linear operator T on a Banach space is the complement in the complex plane \mathbb{C} of the spectrum \sigma(T), so \rho(T) = \mathbb{C} \setminus \sigma(T).^[13] The spectrum \sigma(T) is always a nonempty, compact subset of \mathbb{C}, hence closed and bounded.^[14] Specifically, \sigma(T) is contained in the closed disk \{ \lambda \in \mathbb{C} : |\lambda| \leq \|T\| \}, ensuring that all sufficiently large |\lambda| belong to \rho(T).^[15] The resolvent set \rho(T) is an open subset of \mathbb{C}.^[13] To see this, if \lambda_0 \in \rho(T), then the resolvent operator R(\lambda_0, T) = (\lambda_0 I - T)^{-1} exists and is bounded; for perturbations \lambda = \lambda_0 + \delta with |\delta| < 1 / \|R(\lambda_0, T)\|, the Neumann series \sum_{n=0}^\infty (\delta R(\lambda_0, T))^n R(\lambda_0, T) converges to show that \lambda \in \rho(T), so a disk around \lambda_0 lies in \rho(T).^[14] The resolvent set exhibits translation invariance under scalar shifts of the operator: for any scalar c \in \mathbb{C}, \rho(T + cI) = \rho(T) + c.^[1] Similarly, the resolvent set is invariant under similarity transformations: if S is an invertible bounded linear operator, then \rho(S^{-1} T S) = \rho(T).^[1] For a self-adjoint operator T on a Hilbert space, the spectrum \sigma(T) is a nonempty closed subset of the real line \mathbb{R}, so \rho(T) = \mathbb{C} \setminus \sigma(T) consists of all nonreal complex numbers together with the real numbers outside \sigma(T).^[16]

Analytic Properties of the Resolvent

The resolvent function R(\cdot, T): \rho(T) \to \mathcal{B}(X), defined for a closed linear operator T on a complex Banach space X, is holomorphic on the resolvent set \rho(T).^[17] This holomorphy follows from the local representation of the resolvent via the Neumann series expansion around points in \rho(T), which converges uniformly on compact subsets and yields a power series in (\lambda - \mu) for \mu \in \rho(T).^[12] The derivative of the resolvent satisfies the resolvent equation

\frac{d}{d\lambda} R(\lambda, T) = -R(\lambda, T)^2,

which can be verified by differentiating the defining identity (\lambda I - T) R(\lambda, T) = I with respect to \lambda, using the product rule and the fact that T commutes with R(\lambda, T).^[17] Higher derivatives exist and are given recursively by \frac{d^n}{d\lambda^n} R(\lambda, T) = (-1)^n n! R(\lambda, T)^{n+1}.^[12] Points in the spectrum \sigma(T) correspond to singularities of the resolvent function R(\cdot, T), which are either poles or essential singularities.^[17] For an isolated point \mu \in \sigma(T), if \mu is a pole of finite order, the resolvent admits a Laurent series expansion in a punctured disk around \mu:

R(\lambda, T) = \sum_{n=-\infty}^{\infty} a_n (\lambda - \mu)^n,

where the principal part \sum_{n=-m}^{-1} a_n (\lambda - \mu)^n (with m < \infty) determines the order of the pole, and the coefficients a_n for n \geq 0 provide the regular part via the reduced resolvent.^[17] The residue at a simple pole is the spectral projection onto the corresponding eigenspace, while essential singularities arise at points of the continuous or residual spectrum without isolated character.^[12] This expansion facilitates the analytic continuation of the resolvent across isolated spectral points and underpins perturbation analyses near eigenvalues. Runge's approximation theorem extends to the operator-valued setting, allowing uniform approximation of the resolvent R(\lambda, T) on compact subsets of \rho(T) by rational functions in \lambda with poles restricted to \sigma(T).^[18] Specifically, for a compact set K \subset \rho(T) whose complement is connected and contains \sigma(T), there exist rational operator functions r_k(\lambda) (polynomials in numerator and denominator) such that \| r_k(\lambda) - R(\lambda, T) \| \to 0 uniformly for \lambda \in K, with poles of r_k approaching points in \sigma(T). This is derived by applying Runge's theorem componentwise in the holomorphic functional calculus, ensuring the approximants respect the operator structure.^[18] For compact operators T \in \mathcal{B}(X), the spectrum \sigma(T) consists of at most countably many eigenvalues, with 0 as the only possible accumulation point.^[19] Consequently, the resolvent set \rho(T) = \mathbb{C} \setminus \sigma(T) is connected and open, excluding a discrete set accumulating solely at 0, which enhances the global holomorphy of R(\cdot, T) outside this accumulation and supports uniform bounds on large |\lambda|.^[19] This structure implies that non-zero spectral points are isolated poles of R(\cdot, T), with residues being finite-rank projections.^[12]

Relation to Spectrum and Applications

Complement to the Spectrum

The resolvent set \rho(T) of a linear operator T on a Banach space is the complement of the spectrum \sigma(T) in the complex plane \mathbb{C}. The spectrum \sigma(T) is always closed, making \rho(T) open by definition.^[12] For bounded operators, \sigma(T) is compact and contained within the closed disk \{ \lambda \in \mathbb{C} : |\lambda| \leq \|T\| \}, ensuring that \rho(T) is dense in \mathbb{C}.^[12] The connected components of \rho(T) are instrumental in spectral decompositions, as they allow the underlying space to be decomposed into spectral subspaces corresponding to isolated parts of the spectrum separated by contours in \rho(T). In the case of normal operators on Hilbert spaces, the spectrum satisfies \sigma(T) \subset \{ \lambda \in \mathbb{C} : |\lambda| \leq \|T\| \}, with the spectral radius equaling the operator norm \|T\|.^[20] Here, \rho(T) possesses exactly one unbounded connected component, which is the exterior of the disk containing \sigma(T), facilitating the analysis of behavior at infinity. This structure underscores the topological interplay between \rho(T) and \sigma(T), where the openness and density of the resolvent set enable the extension of holomorphic functions across large regions of the plane. A key application of this complement arises in the Dunford-Schwartz functional calculus, which defines functions of the operator via contour integrals over paths lying entirely in \rho(T). For a function f holomorphic in a neighborhood of \sigma(T), the operator f(T) is given by

f(T) = \frac{1}{2\pi i} \int_\Gamma f(\lambda) R(\lambda, T) \, d\lambda,

where \Gamma is a closed contour in \rho(T) enclosing \sigma(T). Such integrals exploit the analytic properties of the resolvent to represent spectral projections and functional powers, with the choice of contour determined by the connected components of \rho(T). The boundary of the resolvent set coincides with that of the spectrum, \partial \rho(T) = \partial \sigma(T), reflecting their complementary topology in \mathbb{C}. Points on this boundary often belong to the continuous or residual spectrum and serve as approximate eigenvalues, meaning for any \lambda \in \partial \sigma(T), there exists a sequence of unit vectors x_n such that \|(T - \lambda I)x_n\| \to 0.^[12] This boundary behavior highlights the transition from invertibility in \rho(T) to singular perturbations near \sigma(T).

Role in Spectral Theory

The resolvent set plays a pivotal role in spectral theory by facilitating the decomposition of operators into spectral components and enabling the construction of functional calculi. Introduced by David Hilbert in the early 1900s through his analysis of linear integral equations, the resolvent kernel provided a means to express solutions as expansions in terms of eigenvalues and eigenfunctions for symmetric kernels.^[21] This laid foundational groundwork for understanding operator spectra in infinite-dimensional spaces. The concept was formalized and extended by Frigyes Riesz during the 1910s to 1930s, particularly in his development of spectral theory for compact operators on Hilbert spaces, where the resolvent helped establish completeness of eigenfunction systems.^[22] A key application arises in the spectral theorem for self-adjoint operators, where the resolvent integral representation underpins the functional calculus. For a self-adjoint operator T on a Hilbert space X, and a suitable function f, the operator f(T) can be defined as

f(T) = \frac{1}{2\pi i} \int_{\Gamma} f(\lambda) R(\lambda, T) \, d\lambda,

where \Gamma is a closed contour enclosing the spectrum \sigma(T) but lying in the resolvent set \rho(T).^[23] This integral decomposes X into spectral subspaces corresponding to Borel sets in \sigma(T), allowing the resolution of the identity and the representation of T as a multiplication operator in a suitable basis.^[24] Such constructions are essential for applying spectral theory to differential operators and quantum mechanical Hamiltonians. Illustrative examples highlight the diversity of resolvent sets in concrete settings. For finite matrices representing bounded operators on finite-dimensional spaces, \rho(T) = \mathbb{C} \setminus \{\text{eigenvalues of } T\}, reflecting the discrete nature of the spectrum.^[25] In the case of the bilateral shift operator on \ell^2(\mathbb{Z}), the spectrum \sigma(T) is the unit circle, with \rho(T) comprising the exterior of the unit disk (and interior, though the exterior dominates resolvent estimates).^[26] For the unbounded Laplacian -\Delta on L^2(\mathbb{R}), the spectrum is the continuous half-line [0, \infty), so \rho(T) = \mathbb{C} \setminus [0, \infty), determined via Fourier transform.^[24] The framework extends naturally to unbounded operators that are closed and densely defined on a Hilbert space, where the resolvent set \rho(T) is defined analogously as the values \lambda \in \mathbb{C} for which \lambda - T: \dom(T) \to X is bijective with bounded inverse extending to all of X.^[27] In this setting, \rho(T) is maximal among possible extensions, and the spectrum \sigma(T) is contained in the closure of the numerical range W(T) = \{\langle Tx, x \rangle : x \in \dom(T), \|x\|=1\}, providing bounds on spectral location for operators like sectorial ones in evolution equations.^[28]

References

[1]
[PDF] Spectrum (functional analysis)
Mar 12, 2013 · Definition. Let be a a bounded linear operator acting on a Banach space over the scalar field , and be the identity operator on . The ...
[2]
[PDF] Level sets of the resolvent norm of a linear operator revisited - arXiv
Apr 10, 2015 · An example of a bounded linear operator on a Banach space for which the resolvent norm is constant in a neighbourhood of zero was constructed in ...
[3]
Banach Space -- from Wolfram MathWorld
A Banach space is a complete vector space with a norm . Two norms and are called equivalent if they give the same topology, which is equivalent to the ...
[4]
[PDF] Banach Spaces - UC Davis Mathematics
Definition 5.1 A Banach space is a normed linear space that is a complete metric space with respect to the metric derived from its norm.
[5]
Banach Space - an overview | ScienceDirect Topics
A Banach space is a normed vector space that is complete with respect to the norm topology (meaning that the limit of any sequence of vectors is itself ...
[6]
Bounded Linear Operator - an overview | ScienceDirect Topics
Linear operators acting between normed linear spaces that are continuous with respect to the norms are called bounded.
[7]
[PDF] Engr210a Lecture 6: Linear analysis and systems
L(V,Z) is the set of all bounded linear operators mapping V to Z. • L(V) is ... The set of linear operators on any Banach space V forms a Banach algebra.
[8]
[PDF] Etudes of the resolvent.pdf
Apr 24, 2020 · In §5.3 we define a solution to the scattering problem and the Jost functions, and we give an explicit formula for the resolvent of the self- ...
[9]
https://arxiv.org/pdf/1606.05993v1
[10]
Banach Algebras and the Spectrum
The following theorem will show that the spectrum is always non-empty, and ... non-empty compact subset of C. Moreover, the spectral radius satisfies.
[11]
Non-emptiness of spectrum σ(a) in non-Archimedean Banach ...
Dec 27, 2021 · The standard proof involves showing that if σ(a) is empty, then for each ψ∈A∗, the map λ↦ψ((λ1A−a)−1):C→C is bounded, entire and vanishes at ...
[12]
[PDF] The Resolvent of an Operator - UW Math Department
Page 7. Resolvent set of T ∈ B(X) Definition If T ∈ B(X), the resolvent set ρ(T) ⊂ C is set of z such that (zI − T) is invertible. Let RT (z)=(zI − T)−1 for z ...
[13]
[PDF] 1. Spectral theory of bounded self-adjoint operators In the essential ...
resolvent set of T if the operator T − zI is one-to-one and onto, i.e., is invertible on H. By the open mapping theorem the operator. (T − zI)−1 : H→H is ...
[14]
[PDF] Chapter 9: The Spectrum of Bounded Linear Operators
(A). If A - X H is one-to-one and onto, then the open mapping theorem implies that ... C ‡ [0, 1] is in the resolvent set of ڈ . If X [0, 1], thenDڈ - X H is not ...
[15]
[PDF] 13 Spectral theory
The spectrum of a bounded linear operatorT is compact, and in particular bounded by its norm: σ(T) ⊂ B∥T ∥(0). Proof: Suppose |λ| > ∥T ∥. Then I − λ. −1. T is ...
[16]
[PDF] Class notes, Functional Analysis 7212 - OSU Math
Apr 1, 2019 · The resolvent set ρ(T) of a a densely defined operator T : D → X is defined as the set ρ(T) of λ ∈ C s.t.. (T − λ) is injective from D to ...<|control11|><|separator|>
[17]
[PDF] Functional Analysis Princeton University MAT520 Lecture Notes
Aug 18, 2023 · Functional Analysis. Graduate Studies in Mathematics. American Mathematical. Society, 2018. [Con19] J.B. Conway. A Course in Functional Analysis ...
[18]
[PDF] Functional Analysis - Univr
The resolvent set is open in C. More- over, if |λ| > kTk then {λ ∈ C, |λ| > ||T||} ⊂ ρ(T). Actually, denoting r(T) = lim supn(||Tn||)1/n ≤ ||T|| the ...
[19]
[PDF] Invariant subspaces for invertible operators on Banach spaces
We define the resolvent set of T as the set ρ(T) of scalars λ ∈ K such that λ − T is invertible, this is, the operator. R(λ, T)=(λ − T)−1 exists and it ...
[20]
[PDF] 13 Spectral theory
Spectral theory aims to generalize diagonalization for operators on Hilbert space, defining the spectrum as where T-λI is not invertible with bounded inverse.
[21]
[PDF] Perturbation Theory
Page 1. C L A S S I C S I N M A T H E M AT I C S. Tosio Kato. Perturbation Theory for Linear Operators ... functional analysis. But since the book is partly ...<|control11|><|separator|>
[22]
https://pages.stern.nyu.edu/~dbackus/BCZ/HS/Harrell_operatortheory_04.pdf
[23]
[PDF] Spectral Theory for Compact Self–Adjoint Operators
The spectrum of L, σ(L), is defined as the complement of the resolvent set: σ(L) := ρ(L)С. This agrees with the definition of the spectrum in the matrix case, ...<|control11|><|separator|>
[24]
[PDF] 7. Operator Theory on Hilbert spaces - KSU Math
Corollary 7.2 (Spectral Radius Formula for normal operators). Let H be a. Hilbert space, and let T ∈ B(H) be a normal operator. Then one has the equality.<|control11|><|separator|>
[25]
[PDF] On the origin and early history of functional analysis - DiVA portal
In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations ...
[26]
[PDF] A Short History of Operator Theory - NYU Stern
In 1916 Riesz created the theory of what he called "completely continuous" operators, now more familiarly compact operators. Since he wrote this in Hungarian, ...
[27]
[PDF] Chen,Aden.pdf - UChicago Math
With the spectral theorem, we can now construct a functional calculus for self-adjoint operators. For a bounded Borel function 𝑓 ∈ bb(R) : R → C, we define.<|control11|><|separator|>
[28]
[PDF] Spectral theory in Hilbert spaces (ETH Zürich, FS 09) E. Kowalski
We have already observed that bounded operators can lead to unbounded ones if one considers the resolvent (T − λ)−1 for λ in the continuous spectrum. This ...
[29]
[PDF] Spectral Theory Examples
Sep 27, 2018 · Example 2 (Spectrum of Shift Operators) Define the right and left shift operators acting on ℓ2 by. L(α1,α2,α3, ···)=(α2,α3, ···). R(α1,α2,α3 ...
[30]
[PDF] A Guide To Spectral Theory | HAL
The resolvent set ρ(T) of T is the set of all z ∈ C such that T − z : Dom (T) → H is bijective. Note that, by the closed graph theorem ...
[31]
[PDF] NOTES ON THE NUMERICAL RANGE - Michigan State University
If T is a bounded linear operator on a Hilbert space H, then the spectrum of T is contained in the closure of the numerical range of T. Proof. Because both ...Missing: disk | Show results with:disk