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Functional calculus

In mathematics, particularly within and , functional calculus refers to a collection of techniques that extend the application of scalar functions to linear s, allowing the definition of f(T) for a function f and an T on a , typically via the and properties of the operator's . This framework builds on the functional calculus, where polynomials in T are straightforward, and generalizes it to broader classes of functions such as continuous, analytic, or measurable ones, ensuring the resulting inherits key algebraic and topological properties like boundedness and normality. The continuous functional calculus applies to normal operators T on a H, defining f(T) for continuous functions f on the \sigma(T) through the f(T) = \int_{\sigma(T)} f(\lambda) \, dE(\lambda), where E is the of T; this preserves the C^*- structure and is fundamental for . For broader applicability, the extends this to all bounded Borel-measurable functions on \sigma(T), using a unique spectral measure P on the Borel \sigma- such that f(T) = \int_{\sigma(T)} f \, dP, enabling the treatment of discontinuous functions while maintaining a *- from the of bounded measurable functions to the bounded operators on H. This construction is unique for normal operators and plays a crucial role in determining eigenvalues (via P(\{\lambda\}) \neq 0) and characterizing classes like unitary operators (where \sigma(T) \subset \partial \mathbb{D}). In non-commutative settings, such as Banach algebras, the analytic functional calculus defines f(T) for holomorphic functions f on a neighborhood of the using integrals, f(T) = \frac{1}{2\pi i} \oint_C f(z) (z - T)^{-1} \, dz, where C encloses \sigma(T); this is essential for sectorial operators and unbounded closed operators with non-empty resolvent sets, facilitating applications in evolution equations and semigroup theory. Functional calculus has profound implications across , where it underpins the definition of observables as operators and their functions, and in partial differential equations, for analyzing elliptic and operators through methods. Extensions to multivariable or quaternionic settings further broaden its scope, though the classical univariate case remains central to operator algebras.

Fundamentals

Definition and basic concepts

In , a functional calculus provides a framework for applying functions to linear in a way that extends familiar algebraic operations from scalar to operators. Specifically, for a given linear operator A acting on a Banach or , a functional calculus assigns to each f in an appropriate class of functions (typically defined on a containing the of A) a corresponding operator f(A), such that the assignment preserves algebraic structures, including f(A) + g(A) = (f + g)(A) and f(A) g(A) = (f g)(A) for compatible functions f and g. This construction is an homomorphism, ensuring that the resulting operators behave analogously to compositions of scalar functions. Central to functional calculus are linear operators on Banach or Hilbert spaces, which are complete normed vector spaces over the numbers. Operators are classified as bounded if they map the entire to itself with a finite , meaning \|A x\| \leq M \|x\| for some constant M and all x in the ; unbounded operators, in contrast, are defined only on a dense (their ) and may not extend continuously to the whole . A key prerequisite is the of an A, denoted \sigma(A), which is the set of numbers \lambda \in \mathbb{C} such that A - \lambda I (where I is the ) is not invertible in the of bounded operators. The plays a crucial role, as functional calculi are typically defined for functions analytic or continuous on regions containing \sigma(A), and it determines the possible values where the behaves pathologically. Particularly important are self-adjoint and normal operators on s, which are equipped with an inner product inducing a norm. A densely defined operator A on a H is if it equals its A^*, meaning \operatorname{dom}(A) = \operatorname{dom}(A^*) and \langle A x, y \rangle = \langle x, A^* y \rangle for all x \in \operatorname{dom}(A), y \in \operatorname{dom}(A^*). An operator is normal if it commutes with its , A A^* = A^* A. These operators are foundational in because their spectra lie on the real line (for ) or satisfy additional symmetry, allowing decomposition via the into measures supported on the . The simplest functional calculus arises for polynomials, providing an explicit starting point. For a polynomial p(z) = \sum_{k=0}^n c_k z^k and a linear operator A (bounded or unbounded on a suitable domain), the polynomial functional calculus defines p(A) = \sum_{k=0}^n c_k A^k, where A^0 = I and higher powers are composed iteratively, with the domain of p(A) taken as \operatorname{dom}(A^n) to ensure well-definedness. For simple cases like the exponential function, which can be viewed as the limit of its power series \exp(z) = \sum_{k=0}^\infty \frac{z^k}{k!}, the construction extends analogously when the series converges in the operator norm, yielding \exp(A) = \sum_{k=0}^\infty \frac{A^k}{k!}.

Historical development

The concept of functional calculus originated in the 18th and 19th centuries as a framework for the , where mathematicians like Leonhard Euler and developed methods to optimize functionals—mappings from functions to scalars—such as those arising in problems of minimal surfaces or geodesics. This early usage treated functionals as higher-order analogs of ordinary functions, with Euler's work in the 1760s laying foundational principles for variational principles in . By the late , advanced this into a systematic "functional calculus" through his studies of functionals and their derivatives, published in a series of papers starting in the 1880s, which introduced concepts like the Gâteaux derivative for infinite-dimensional spaces. In the early , the term began to diverge from its variational roots toward , influenced by the study of integral equations. David Hilbert's foundational work from 1904 to 1910 on linear integral equations established key ideas in for operators on L² spaces, providing the groundwork for applying functions to infinite-dimensional operators through expansions in eigenfunctions. This shift was pivotal, as Hilbert's approach to operators and continuous spectra in his 1906 papers bridged finite theory to infinite dimensions. extended these ideas in the 1920s and 1930s, developing the for normal operators and introducing functional calculus for unbounded operators in . By the early 20th century, functional calculus was formalized within , with Marshall Stone's 1932 book Linear Transformations in Hilbert Space and Their Applications to providing a comprehensive of spectra and resolutions of the identity, solidifying its distinction from variational methods. This separation became clear post-, as -theoretic applications in quantum physics and overshadowed earlier variational contexts. In the modern era, integrated functional calculus into the theory of C*-algebras during the and , with his 1941 paper on the formula in normed algebras enabling representations of via continuous functions on their spectra. These developments, building on the as a core tool, established functional calculus as a cornerstone of algebras.

Polynomial functional calculus

For finite-dimensional operators

In finite-dimensional spaces, the polynomial functional calculus applies directly to matrices, providing a concrete and computable framework. For an n \times n complex matrix A, given a polynomial p(z) = \sum_{k=0}^m c_k z^k, the operator p(A) is defined as p(A) = \sum_{k=0}^m c_k A^k, where A^0 = I_n is the identity matrix and higher powers are computed via matrix multiplication. This construction yields a unital algebra homomorphism from the ring of polynomials \mathbb{C} to the matrix algebra \mathbb{C}^{n \times n}, preserving addition and multiplication: (p + q)(A) = p(A) + q(A) and (pq)(A) = p(A) q(A). Key properties follow from this homomorphism. Notably, p(A) commutes with A, since each power A^k satisfies A A^k = A^{k+1} = A^k A. The spectral mapping theorem ensures that the eigenvalues of p(A) are precisely p(\lambda) for each eigenvalue \lambda of A, with algebraic multiplicities preserved in the characteristic polynomial. Additionally, the minimal polynomial m_A(z) of A, the monic polynomial of least degree such that m_A(A) = 0, divides any other annihilating polynomial q(z) with q(A) = 0. The Cayley-Hamilton theorem implies that the characteristic polynomial \chi_A(z) annihilates A, so \chi_A(A) = 0; this relation allows reduction of higher-degree terms in computations, as powers A^k for k \geq n can be expressed linearly in terms of lower powers using the coefficients of \chi_A. For diagonalizable matrices, the calculus simplifies further. If A = P D P^{-1} with D = \operatorname{diag}(\lambda_1, \dots, \lambda_n), then p(A) = P \, p(D) \, P^{-1}, where p(D) is the \operatorname{diag}(p(\lambda_1), \dots, p(\lambda_n)). Consider the explicit example of a 2×2 A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, which has eigenvalues e^{i\theta} and e^{-i\theta} and is diagonalizable over \mathbb{C}. For the monomial p(z) = z^k, p(A) = A^k represents a by k\theta, as A^k = \begin{pmatrix} \cos(k\theta) & -\sin(k\theta) \\ \sin(k\theta) & \cos(k\theta) \end{pmatrix}, illustrating how the calculus preserves the rotational structure for powers. Algorithmically, the minimal polynomial can be computed using companion matrices, which arise in the rational canonical form of A. Form the Krylov matrices from the powers I_n, A, \dots, A^{n-1} and identify linear dependencies; the companion matrix of a candidate polynomial encodes the action on a cyclic subspace, allowing verification of annihilation and determination of invariant factors that yield m_A. This approach leverages the fact that if C is the companion matrix of a monic polynomial q(z) of degree n, then both the characteristic and minimal polynomials of C are q(z).

For infinite-dimensional operators

In the infinite-dimensional setting, the polynomial functional calculus extends to bounded linear operators on a Hilbert space H. For a bounded operator A \in B(H) and a polynomial p(z) = \sum_{k=0}^n c_k z^k, the operator p(A) is defined as p(A) = \sum_{k=0}^n c_k A^k, where the series converges in the operator norm topology due to the boundedness of A. The unital *-algebra generated by \{I, A, A^2, \dots \} consists of all finite linear combinations of these powers and is the algebraic of such terms; its in the yields the closed of B(H) comprising the norm limits of polynomials in A. For a A, each p(A) remains compact, and the polynomials provide finite-rank approximations when A admits a finite-rank structure, facilitating through truncation to finite-dimensional subspaces. A notable limitation arises in infinite dimensions, where the structure of the generated algebra can differ markedly from the finite-dimensional case. Consider the unilateral S on \ell^2(\mathbb{N}), defined by S(e_n) = e_{n+1} for the standard \{e_n\}, which is an with \|S\| = 1. The C*-algebra generated by polynomials in S coincides with the full Toeplitz algebra \mathcal{T} = C^*(S), a non-commutative containing all compact perturbations of Toeplitz operators with continuous symbols on the unit circle; unlike the finite-dimensional scenario, where invariant subspaces yield non-trivial ideals corresponding to direct sum decompositions, the Toeplitz algebra has only trivial ideals, reflecting the absence of non-trivial reducing subspaces for S. In infinite dimensions, the minimal polynomial of an operator may not exist, as the annihilator ideal in the polynomial ring can be trivial—for instance, no non-zero polynomial annihilates the unilateral shift S, since its powers remain isometries without nilpotent behavior. Consequently, the minimal polynomial fails to provide the invariant subspace decomposition available in finite dimensions; instead, one relies on spectral subspaces derived from the spectral measure to decompose the space. For a A, the operator norm satisfies \|p(A)\| = \sup_{\lambda \in \sigma(A)} |p(\lambda)|, which aligns the norm with the supremum norm on the spectrum and connects to the spectral radius formula r(A) = \|A\| via the of the generated .

Spectral functional calculus

Borel functional calculus

The Borel functional calculus provides a framework for applying Borel-measurable functions to normal operators on a , extending the scope beyond polynomials or continuous functions by leveraging the structure of spectral measures. It presupposes the for normal operators, which asserts that for a bounded normal operator A on a separable H, there exists a unique projection-valued measure E (the spectral measure) defined on the Borel \sigma-algebra \mathcal{B}(\mathbb{C}) such that A = \int_{\mathbb{C}} \lambda \, dE(\lambda) in the strong operator topology, with the support of E contained in the spectrum \sigma(A) \subset \mathbb{C}. This resolution of the identity allows the operator to be decomposed into its spectral components, enabling the definition of functions of A through integration against E. The construction of the proceeds by first defining it for simple functions—finite linear combinations of characteristic functions of Borel sets—and then extending by density and continuity. Specifically, for a bounded Borel-measurable function f: \mathbb{C} \to \mathbb{C} such that f vanishes outside \sigma(A), the f(A) is given by f(A) = \int_{\mathbb{C}} f(\lambda) \, dE(\lambda), where the is interpreted in the strong operator sense: for simple functions, it is a finite [sum \sum](/page/Sum_Sum) c_k E(B_k), and for general bounded Borel f, it is the strong limit of integrals over simple function approximations converging pointwise \mu- for every finite Borel measure \mu induced by vectors in H. This yields a unique bounded f(A) \in B(H) that is an isometric unital *-homomorphism from the algebra of bounded Borel functions on \sigma(A) (modulo almost everywhere equivalence) to the C*-algebra generated by A. The polynomial functional calculus arises as a special instance, via approximation of polynomials by s. Key properties follow directly from the spectral integration. If f is real-valued, then f(A) is self-adjoint, as E is a positive measure and the integral preserves the adjoint. The operator norm satisfies \|f(A)\| \leq \|f\|_\infty = \sup_{\lambda \in \sigma(A)} |f(\lambda)|, with equality if f is continuous, reflecting the isometric nature of the map. Composition is resolved functionally: for bounded Borel f, g with fg defined, f(A) g(A) = (fg)(A), ensuring the calculus respects algebraic structure. For any \psi \in H, the expectation satisfies \langle \psi, f(A) \psi \rangle = \int_{\mathbb{C}} f(\lambda) \, d\langle \psi, E(\cdot) \psi \rangle (\lambda), where \langle \psi, E(\cdot) \psi \rangle is the scalar spectral measure induced by \psi, a finite positive Borel measure on \mathbb{C} with total mass \|\psi\|^2. This formula underscores the probabilistic interpretation in applications like quantum mechanics. A concrete example illustrates the calculus: consider the multiplication operator A on L^2(\mathbb{R}, d\mu) for a \sigma-finite measure \mu, defined by (A \phi)(x) = x \phi(x) for \phi \in L^2(\mathbb{R}, d\mu), which is a self-adjoint normal operator. Here, A has spectrum \sigma(A) equal to the essential range of the identity function x with respect to \mu, and the spectral measure is E(B) \phi = \chi_B(x) \phi(x) for Borel B \subset \mathbb{R}, where \chi_B is the characteristic function. Then f(A) is the multiplication operator by f(x), i.e., (f(A) \phi)(x) = f(x) \phi(x), provided f is bounded Borel-measurable, directly realizing the integral \int f(\lambda) \, dE(\lambda). This model underpins many concrete computations in functional analysis.

Continuous functional calculus

The continuous functional calculus provides a method to associate continuous functions on the of a element in a with elements of the algebra itself, extending the functional calculus in a unique and structure-preserving way. For a element a in a unital A, let \sigma(a) \subset \mathbb{C} denote the of a, which is a compact . The calculus defines a map \varphi: C(\sigma(a)) \to C^*(a), where C^*(a) is the unital generated by a, such that \varphi(f) = f(a) for f \in C(\sigma(a)). This map is a unital isometric *-homomorphism, uniquely determined as the continuous extension of the functional calculus on a. Key properties of this construction include the preservation of the and the . Specifically, the satisfies \sigma(f(a)) = f(\sigma(a)) for any f \in C(\sigma(a)), ensuring that the functional calculus respects the spectral structure of a. Additionally, the norm is preserved via \|f(a)\| = \|f\|_\infty, where \|f\|_\infty = \sup_{\lambda \in \sigma(a)} |f(\lambda)|, reflecting the isometric nature of \varphi. These properties arise from the isomorphism between C(\sigma(a)) and C^*(a), which aligns with the broader framework of spectral theory in operator algebras. In the commutative case, the continuous functional calculus relies on the Gelfand transform, which represents the A as continuous functions on its \Omega(A), the space of nonzero homomorphisms from A to \mathbb{C}. For a normal a \in A, f(a) corresponds to the multiplication operator by f \circ \hat{a} on C(\Omega(A)), where \hat{a} is the Gelfand transform of a. This applies particularly to normal operators on Hilbert spaces, where the C*-algebra of bounded operators provides a concrete realization. A representative example occurs with unitary operators, which are normal elements with spectrum on the unit circle \mathbb{T}. For a unitary U in a C*-algebra, the continuous functional calculus defines f(U) for any f \in C(\mathbb{T}), and if |f(z)| = 1 for all z \in \mathbb{T}, then f(U) is also unitary, preserving the group structure under multiplication. This construction is foundational for analyzing functions of unitaries in and .

Holomorphic functional calculus

For bounded operators

The holomorphic functional calculus for bounded operators relies on the resolvent function, which provides the foundational tool for extending to operators. For a bounded linear operator A acting on a complex Banach space X, the resolvent is defined as R(\zeta, A) = (\zeta I - A)^{-1} for \zeta \in \mathbb{C} \setminus \sigma(A), where \sigma(A) denotes the spectrum of A. This resolvent exists as a bounded operator precisely when \zeta lies outside \sigma(A), and it is holomorphic as a function of \zeta in that region. Given a function f that is holomorphic in an open neighborhood \Omega of \sigma(A), the functional calculus constructs f(A) via a contour integral over a suitable closed curve \Gamma in \Omega that encloses \sigma(A) and winds once positively around it. Specifically, f(A) = \frac{1}{2\pi i} \int_\Gamma f(\zeta) R(\zeta, A) \, d\zeta. This integral defines a bounded linear operator on X, independent of the choice of \Gamma as long as it satisfies the conditions. The construction, known as the Dunford integral or Riesz-Dunford calculus, ensures that f(A) is well-defined and captures the action of f on the spectral data of A. The mapping f \mapsto f(A) exhibits key analytic properties: it is holomorphic in the sense that if f_n \to f uniformly on compact subsets of \Omega, then f_n(A) \to f(A) in the . Moreover, f(A) commutes with A, i.e., [f(A), A] = 0, and the norm satisfies \|f(A)\| \leq \frac{\ell(\Gamma)}{2\pi} \max_{\zeta \in \Gamma} |f(\zeta)| \max_{\zeta \in \Gamma} \|R(\zeta, A)\|, where \ell(\Gamma) is the length of \Gamma; a refined estimate bounds \|f(A)\| \leq \max_{\zeta \in \Gamma} |f(\zeta)| / d(\Gamma, \sigma(A)) under suitable contour choices, with d(\Gamma, \sigma(A)) the minimum distance from \Gamma to \sigma(A). For monomials, the recovers powers: if f(z) = z^n, then f(A) = A^n, as follows from applied to the resolvent. This integral representation forms the core of the Dunford-Schwartz for bounded operators. A representative example arises with the f(z) = e^z, applicable when \sigma(A) lies in the open left half-plane \{z : \Re z < 0\}. Here, e^{tA} for t \geq 0 generates a contraction semigroup, satisfying the abstract Cauchy problem for evolution equations, with the contour integral ensuring stability and analytic continuation properties.

For unbounded operators

The holomorphic functional calculus for unbounded operators is primarily developed for sectorial operators, which are densely defined closed operators A on a Banach space whose spectrum \sigma(A) is contained in a sector \Sigma_\theta = \{ z \in \mathbb{C} \setminus \{0\} : |\arg z| \leq \theta \} for some \theta \in [0, \pi), with the resolvent R(\lambda, A) = (\lambda I - A)^{-1} satisfying \|R(\lambda, A)\| \leq M / |\lambda| for all \lambda outside a larger sector \Sigma_\phi with \phi > \theta and some constant M \geq 1. For a f holomorphic in a sector \Sigma_{\theta + \varepsilon} with \varepsilon > 0, the operator f(A) is constructed via the f(A) = \frac{1}{2\pi i} \int_\Gamma f(\zeta) R(\zeta, A) \, d\zeta, where \Gamma is a suitable in \mathbb{C} \setminus \overline{\Sigma_\theta} that encloses \sigma(A), ensuring the integral converges due to the bounded resolvent outside the sector. This extends the calculus from bounded operators, which correspond to the special case of a sector of 0. The resulting f(A) is generally an unbounded operator, particularly when f exhibits growth at infinity, and is defined on a dense domain D(f(A)) that contains D(A) under appropriate conditions on f. For well-posedness, A must be sectorial or m-accretive (meaning A is accretive and I + tA is surjective for t > 0), with the sector angle determining the angle of the generated analytic semigroup; specifically, if the sectoriality angle is less than \pi/2, A generates a bounded analytic semigroup on a sector of angle \pi/2 - \theta. In particular, for f(z) = e^{-z}, the operator f(A) generates an analytic semigroup, facilitating the study of evolution equations. A key norm estimate for the boundedness of f(A) is given by \|f(A)\| \leq M \sup_{|\arg \zeta| < \theta + \varepsilon} \frac{|f(\zeta)|}{|\zeta|}, where the supremum is taken over the enlarged sector and M depends on the resolvent bound, providing growth control essential for applications. This bound ensures the calculus is consistent with the semigroup properties and extends to functions in H^\infty(\Sigma_{\theta + \varepsilon}). A representative example is the negative Laplacian A = -\Delta on L^2(\mathbb{R}^n) with domain H^2(\mathbb{R}^n), which is sectorial of angle 0 (self-adjoint and positive) and admits a holomorphic calculus. For f(z) = z^{-\alpha/2} with $0 < \alpha < 2, the operator f(A) yields the fractional power (-\Delta)^{-\alpha/2}, corresponding to the and used in .

Applications and examples

In quantum mechanics

In quantum mechanics, physical observables such as position, momentum, and energy are represented by self-adjoint operators on a Hilbert space, allowing the application of functional calculus to define functions of these operators, such as energy distributions or probability densities derived from the observable's spectrum. For a self-adjoint operator A, the functional calculus enables the construction of f(A) for suitable functions f, which corresponds to transforming the observable into derived quantities like powers or exponentials that model physical processes. The Borel functional calculus plays a crucial role in associating spectral measures with self-adjoint observables, providing a resolution of the identity through projection-valued measures E(\Delta) for Borel sets \Delta in the spectrum. In the context of quantum measurement, for a state \psi with \|\psi\| = 1, the quantity \langle \psi, E(\Delta) \psi \rangle yields the probability of obtaining a measurement outcome in the interval \Delta, linking the mathematical structure to probabilistic interpretations of quantum outcomes. For time evolution, the holomorphic functional calculus is essential in defining the unitary operator e^{-i t H / \hbar} for a self-adjoint Hamiltonian H, which generates the dynamics of quantum states via the Schrödinger equation. Stone's theorem establishes a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators, ensuring that the generator H is self-adjoint and that the functional calculus applies to produce the time-evolution operator as an exponential function of H. This connection underpins the unitary evolution in quantum dynamics, preserving probabilities and enabling predictions of system behavior over time. A key example arises with the position operator X on the Hilbert space L^2(\mathbb{R}), defined by (X \psi)(x) = x \psi(x), where the functional calculus yields f(X) as multiplication by the function f(x), directly corresponding to position-dependent observables. The momentum operator P = -i \hbar \frac{d}{dx} on the same space has its functional calculus defined via the Fourier transform: for a state \psi, f(P) \psi = \mathcal{F}^{-1} [f(p) \mathcal{F} \psi], where \mathcal{F} denotes the Fourier transform, illustrating how functions of momentum relate to transformations in momentum space.

In partial differential equations

In the context of partial differential equations (PDEs), functional calculus plays a central role in the theory of evolution equations, particularly through the generation of semigroups. For a linear evolution PDE of the form \partial_t u = A u with initial condition u(0) = u_0, where A is an unbounded operator on a Banach space (such as the positive operator -\Delta on a suitable domain), the holomorphic functional calculus for sectorial operators enables the definition of the semigroup e^{tA} as f(A) with f(z) = e^{tz}. This approach leverages the sectoriality of A, meaning its spectrum lies in a left half-plane and the resolvent is bounded appropriately outside a sector, allowing the solution to be expressed as u(t) = e^{tA} u_0. Such semigroups provide mild, strong, or classical solutions depending on the regularity of u_0, and this framework is foundational for analyzing parabolic PDEs like the heat equation. Fractional powers of elliptic operators, defined via functional calculus as A^\alpha = f(A) where f(z) = z^\alpha for \alpha > 0 and sectorial A (e.g., an elliptic with appropriate boundary conditions), are instrumental in formulating and solving nonlocal PDEs. These powers generate subordinate semigroups that model , where the operator A^\alpha replaces the standard Laplacian to capture long-range interactions or memory effects in phenomena like fractional diffusion equations \partial_t u = -A^\alpha u. The holomorphic calculus ensures well-posedness in appropriate function spaces, such as Sobolev spaces, by extending the integer-order theory to non-integer exponents while preserving positivity and maximum principles for positive operators. A key example is the heat equation \partial_t u = \Delta u on \mathbb{R}^n, where A = -\Delta (positive definite) is sectorial, and the semigroup e^{t \Delta} = e^{-t (-\Delta)} is the Gaussian heat kernel, explicitly computable via the Fourier transform as the convolution with (4\pi t)^{-n/2} e^{-|x|^2/(4t)}. Functional calculus facilitates this by defining the resolvent (\lambda - \Delta)^{-1} = (\lambda + (-\Delta))^{-1} explicitly in Fourier space as multiplication by (\lambda + |\xi|^2)^{-1}, from which the semigroup follows via the Laplace transform inversion. For fractional variants like the fractional heat equation \partial_t u = - (-\Delta)^\alpha u with $0 < \alpha < 1, the Balakrishnan formula provides an integral representation for fractional powers: ( \lambda + A )^{-\alpha} v = \frac{\sin(\pi \alpha)}{\pi} \int_0^\infty t^{-\alpha} ( \lambda + t + A )^{-1} (t + A )^{\alpha - 1} v \, dt, enabling the construction of solutions for nonlocal diffusion models. In time-fractional evolution equations of the form {}^C D_t^\alpha u = A u + v(t) (in the Caputo sense), where A is a sectorial spatial operator, the mild solution is given by u(t) = E_{\alpha,1}(t^\alpha A) u_0 + \int_0^t (t-s)^{\alpha-1} E_{\alpha,\alpha}((t-s)^\alpha A) v(s) \, ds, with E_{\alpha,\beta} the . This representation, derived using the holomorphic functional calculus for the generator of the underlying , allows explicit analysis of regularity and asymptotic behavior in applications like or anomalous transport.