Questions tagged [functional-calculus]

Functional calculus allows the evaluation of a function applied to a linear operator or a matrix. The function could be a polynomial, a holomorphic function, a continuous function or a measurable function defined on the spectrum of an operator or a Banach algebra. Functional calculus is a basic and powerful tool in the spectral theory of operators and operator algebras and is part of functional analysis.

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Compactness of a continuous functional calculus of $\mathbf{1}_{\{\lambda\}}$ and normal operator $N$

Consider a normal operator $N$ on complex Hilbert space $\mathcal{H}$, such that for each $\lambda\in\sigma(N)\setminus\{0\}$, $\lambda$ is isolated in $\sigma(N)$ and $\mathrm{dim}\mathrm{ker}(N-\...
Oskar Vavtar's user avatar
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Let $f(x)$ be a continuous function on $[0,1]$ such that $f(1)=0$. $\int_0^1 (f'(x))^2.dx=7$ and $\int_0^1 x^2f(x).dx=\frac13$. Find $\int_0^1f(x).dx$

I have a solution for the above question but i wanted to check if what i am doing is correct and can be done or not. The function i am getting also satisfies both the condition but i still am not sure ...
Yash Shrivastava's user avatar
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derivative of a functional of a functional? [closed]

I am trying to calculate the derivative of the following equation ($P[f]$) w.r.t to $f$ $$P[f] = \int f(s) \times g(Q[f(s)])\,\, ds $$ where $f$ and $g$ are differential functions and $Q$ is a ...
ProbNerd's user avatar
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Is there actually any bijection between characters and global sections of the spectral presheaf?

In pages 5-6 of the article https://arxiv.org/pdf/quant-ph/9911020.pdf, the notion of a spectral presheaf is basically introduced as (in a more contemporany notation): Definition(Valuation): A ...
Felipe Dilho's user avatar
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How can I show that the context generated by an operator $A$ is equal to the set of all Borel measurable functions evaluated at $A$?

If we consider that a context is an unital abelian sub-$C^*$-algebra of some unital $C^*$-algebra $\mathscr{A}$, that agree on the unit, then if we define: Definition: Let $A$ be a unital $∗$-algebra,...
Felipe Dilho's user avatar
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Calculus of Variations no y' in function (?)

I'm a high school student who attempted to use calculus of variations for a project. I have this functional: $$T(\theta)=\int_{0}^{57}\left(\frac{1}{10\cos\left(\theta\right)+w\left(x\right)}+λ\frac{...
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Find the norm of S

Given $C[0,2]$ with the the max-norm. Let $X= \{x\in C ([0,2]): x (1) =0\}$. We define $S:X\to\mathbb{R}$ as $S(x)=\int_{0}^{2}x(t)dt$ . Compute $||S||$. I have already found that $||S||\leq2$, but I ...
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Riesz idempotent (projection) of singleton in $\ell^p$

Edit: After I posted my question, I realised that in this post the same question was asked (and answered). In the meantime useful comments were made under my current post. I've written an answer ...
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Proof of continuous functional calculus via BLT

Background: I'm working on a proof of the spectral theorem as given by Halmos. As I've figured out, the first step, which is omitted in the proof, is to define a continuous functional calculus, i.e. ...
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Range of positive operator

Let $E,F$ be Hilbert $A$-modules, and $T \in \mathcal{L}(E,F)$ with $\lVert T \rVert \leq 1$. Then, $1-T^*T$ has dense range iff $1-TT^*$ has dense range. This is Lemma 10.3 in Lance's book, but I don'...
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Can one reduce the study of unbounded operators to the one of bounded operators?

So I am not very well-versed in functional analysis, but while studying a problem from theoretical solid-state physics I came across the following question: Suppose $\mathcal{H}$ is a complex Hilbert-...
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Why is the Spectrum of an Operator Used as the Domain in Continuous Functional Calculus?

I'm currently working to grasp the concepts of (continuous) functional calculus, aiming to prove the spectral theorem for bounded self-adjoint operators as outlined in "Introduction to Hilbert ...
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How to simultaneously remove a convolution and a multiplication

If you have an equation of the form $$ a(x) \star f(x) + b(x)f(x)=0 $$ Where $\star$ is convolution over the real line and $a,b$ are given functions where you want to solve for the set of possible $f$....
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Quick question: can the marginalization of an operator be seen as the functional calculus of a function?

Since for projectors $\lbrace P_\alpha \rbrace_{\alpha} $ and $\lbrace Q_\beta \rbrace_{\beta} $ on a hilbert space $\mathcal{H}$ that are each a partition of unity: $$\sum_\alpha P_\alpha Q_\beta = ...
Felipe Dilho's user avatar
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functional calculus and laplacian

hi i have a question:the operator $-\Delta$ with domain $H^{2}(R))$ the question is: if $f\in C_{0}^{\infty}(R)$ prove that f(T) is well defined? clearly the answer is by using functional calculus but ...
RIM's user avatar
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how does huber compute the $var(s_n)/E[s_n]$ and $var(d_n)/E[d_n]$?

How does Huber in book 'Robust statistical procedures' in chapter 1 compute the variance of certain statistical functions? He defines the mean square deviation to be $$s_n = \sqrt{\frac{1}{n} \sum \...
PLee's user avatar
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When is the image of the functional calculus of a self-adjoint operator also self-adjoint?

Let $A$ be a self-adjoint operator, either bounded or unbounded, and $f$ a Borel function. Using the functional calculus we may define $f(A)$ as a linear operator. Are there any known conditions on $f$...
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Question on compactness of generator of unitary group

Let $\mathcal{H}$ be complex Hilbert space. Suppose we have a unitary group $\{U_t\},t\in\mathbb{R}$ and by Stone's Theorem we have a unique infinitesimal generator $A:\mathcal{D}(A)\rightarrow\...
Zetton's user avatar
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How to use the functional calculus to prove a bound on the resolvent of a self-adjoint operator

I am following the proof of the following theorem in Reed & Simon's book on functional analysis: Let $\{A_n\}_{n=1}^\infty$ and $A$ be self-adjoint operators and suppose that $A_n \rightarrow A$ ...
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Turning a constrained optimal control problem into unconstrained (Lagrangian)

If I wish to minimize the cost function $$ J(x(\cdot),u(\cdot)) = \int_0^TL(x,u)dt $$ with dynamics constraint $\dot{x}(t) = f(x(t),u(t))$ $\forall t$, many textbooks state that this constrained ...
bsprenger's user avatar
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Interior of set of positive operators

In interior of the set of positive operators the question is about the topology of $\mathcal{P}(\mathcal{H}):=\{A\in \mathcal{L}(\mathcal{H})\mid \langle Ax,x\rangle \geq 0\}$, namely if the interior ...
Paul Thorwarth's user avatar
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Taylor Expansion of Function of Multiple Operators

I would like to Taylor expand a function of multiple operators (which may not commute). For a single operator $A$, I have seen a function $f(A)$ expanded as $$ f(A) = f(0) + A f'(0) + \frac{A^2}{2} f'...
Jack's user avatar
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Maximal $L^p$-regularity of Laplace-Beltrami operator $\Delta$ on closed manifold

It's well-known that the Dirichlet Laplacian $\Delta$ on flat domain is R-sectorial on $\Sigma_{\pi}$ in $L^p$ space for all $p\in (1,\infty)$. I'm wondering if the Laplace-Beltrami operator $\Delta$ ...
celebi's user avatar
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How is the functional calculus explicitly computed for functions that are not analytic?

Let $A$ be a bounded self-adjoint linear operator on a Hilbert space $H$. For any analytic function $f$ whose radius of convergence contains the spectral radius of $A$ we may compute the functional ...
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Find the function $y(x)$ that minimises $ \int_1^2{x^4[y''(x)]^2dx}$

The problem I encounter is that: find the function $y(x)$ with $y(1)=1$ $y'(1)=-2$ $y(2)=1/4$ and $y'(2)=-1/4 $ that minimises $ \int_1^2{x^4[y''(x)]^2dx}.$ I know the equation I should use is $$\...
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Prove that Euler-Lagrange equations remain the same [closed]

I just begin with the Calculus of variation. Here is the problem: Consider the set of n Euler-Lagrange equations for a system with n degrees of freedom: $\frac{\partial f}{\partial y_i}-\frac{d}{dx}\...
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Understanding Proof of Talagrand's Inequality

I am reading Talagrand's seminal paper Concentration of Measure and Isoperimetric Inequalities in Product Spaces. Lemma 2.1.2 on Page 12 obtains the bound $$ \int_\...
Ambar's user avatar
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Strtictly convex functional has unique minimizer on a convex set

On page 8 of the book Regularity of Free Boundaries in Obstacle-type Problems, it considered a functional $$J(u):=\int_D (|\nabla u|^2+2fu)\,dx \qquad D=B(0,1)$$ where $f\in L^\infty(D)$ is fixed. It ...
Szeto's user avatar
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Reed's and Simon's spectral theorem: Why $A\psi=\lambda\psi\implies f(A)\psi=f(\lambda)\psi$?

Let $A$ be a self-adjoint bounded linear operator on a Hilbert space $H$, let $f$ be a bounded Borel measurable function on $\mathbb{R}$ and suppose that $\psi\in H$ such that $A\psi = \lambda\psi$ ...
Epsilon Away's user avatar
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Problem calculating a functional

Let $$J[y]=\int_0^\pi (y')^2dx, \ \ \ \ y(0)=y(\pi)=0$$ which satisfies the additional condition: $$\int_0^\pi y^2dx=1$$ Find the extremals Seemingly this seems easy to solve, but I get a problem ...
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Does the exponential operator $e^{tA}$ defined via functional calculus and via semi group coincide?

Given $X$ is a Hilbert space and consider the operator $A$ on $X$. Functional calculus: given an unbounded operator $A$ (densely defined, closed and self-adjoint), we can define $e^{tA}$ by the ...
celebi's user avatar
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Showing that if $V$ is a closed invariant subspace under $A\in B(H)$ then $V$ is invariant under $f(A)$ for any bounded measurable $f$

Let $H$ be a Hilbert space over the field $\mathbb{C}$ and $A\in B(H)$ be a bounded self-adjoint operator. Let $V\subset H$ be a closed subspace invariant under $A$ and $f:\sigma(A)\to\mathbb{C}$ be a ...
Cartesian Bear's user avatar
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Showing that $||(A-\lambda I)||^{-1}=\frac{1}{d(\lambda,\sigma(A))}$ for $A\in B(H)$ self-adjoint, $\lambda\in\mathbb{C}\setminus\sigma(A)$

Let $A\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator and consider $\lambda\in\mathbb{C}$ outside the spectrum of $A$, i.e., $\lambda\in\mathbb{C}\setminus\sigma(A)$. In this case, $A - \...
Epsilon Away's user avatar
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Fractional representation of fundamental solution

I'm reading Lieb and Frank's paper 'Sharp constants in several inequalities on the Heisenberg group' these days. And I am really confused about the Proposition 4.1 there. They define the 'Laplacian' $\...
IMOS's user avatar
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Understanding the functional derivative

Given a functional $\phi$, it is common in physics texts to see its functional derivative defined as $$\frac{\delta \phi(f)}{\delta f} = \frac{d }{d\epsilon}\phi(f + \epsilon\psi)\Big|_{\epsilon = 0}$$...
CBBAM's user avatar
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Algebra, functional [closed]

this is the problem I've been thinking on for a while: Let $f$ be a functional on $\mathbb{R}^3$ we know that $f(1,2,-1) = 2$ and $kerf = \{(x_1,x_2,x_3) \in \mathbb{R}^3 : 2x_1-x_2+3x_3 = 0\}$ I was ...
weronia_xoxo's user avatar
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Gateaux derivative chain rule varying the argument and the argument of the argument

Suppose I have a functional $G: \mathcal{F} \to \mathbb{R}$ and a functional $F: \mathcal{H} \to \mathbb{R}$. Let $F_\tau = F + \tau \tilde{F}$ and $H_\tau = H + \tau \tilde{H}$ for $F \in \mathcal{F}$...
improbable_probabilist's user avatar
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Does $\det(AB) = \det(A) \det(B)$ hold for differential operators $A$ and $B$?

Following the inconsistency I was facing in a previous post, I was wondering if the following property $$ \det(A) \det(B) = \det(AB) $$ holds for generic $A$ and $B$ differential operators. It looks ...
Adrien Martina's user avatar
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Functional determinant inconsistency

While trying to compute functional determinants, I faced an inconsistency, which I can exemplify by defining the following matrix $$ M = \begin{pmatrix} i \frac{d}{dt} + t & 0 \\\ 0 & i \frac{...
Adrien Martina's user avatar
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Functional derivative of an integral

I don't quite understand the following functional derivative computation when I read a variational inference literature, can someone explain? $$L[q] = E_{q(Y)}[f(Y)]$$ $$\frac{\delta L[q]}{\delta q} = ...
sicheng mao's user avatar
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All spectral projection is trivial implies the operator is identity?

We consider $x\in B(H)$ a self-adjoint operator, then we know that it can determine a unique spectral measure $E$. If we know that $E(\Delta)=0$ or $1$ for all measurable $\Delta\subset \sigma(x)$, ...
Yanyu's user avatar
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2 answers
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Find all real function $f(x)$ satisfy the functional equation [closed]

Find all real function $f(x)$ satisfy this functional equation $|f(x) - f(y) | \leq |\sin(x-y) -x+y| $ for all $x, y \in \mathbb{R}$? My ideas: I showed that the constant functions are a class of ...
ToThichToan's user avatar
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Can functionals mapping from Cartesian products be component-wise decomposed?

Let $(B, \|\cdot\|_B)$ be a Banach space. Then, for any $n\in\mathbb{N},$ $(\boldsymbol{B}=B^n, \|\cdot\|_{\boldsymbol{B}})$ with $\|\boldsymbol{x}\|^2_{\boldsymbol{B}}\,\colon= \sum^n_{i=1}\|x_i\|^...
Obriareos's user avatar
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1 answer
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Integrating the fractional Laplacian over $\Omega$

Let $\Omega$ be a bounded regular open subset of $\mathbb{R}^N$, $N\geq 1$. I want to know if this statement is true : for $u\geq 0$ in $\Omega$, and $u=0$ in $\mathbb{R}^N\setminus \Omega$, $$ \int_\...
Mathslover's user avatar
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Mistake in the Proof for Fundamental lemma of calculus of variation

In this link, the second fundamental lemma of calculus of variation (Lemma 2 in the link): If $\alpha(x)$ is continuous in $[a, b]$, and if $$ \int_a^b \alpha(x) h^{\prime}(x) d x=0 $$ for every ...
user1143399's user avatar
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Response function in weakly coupled differential equations

I have the following system of differential equations: \begin{equation} \frac{d}{dt} \, \begin{pmatrix} x_1 (t) \\ x_2 (t) \\ x_3(t) \\ x_4(t) \\ \vdots \\ x_M (t) \end{pmatrix} = \begin{pmatrix} f_1(...
Saïd M's user avatar
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Is $\max_{x,f} L(x,f) = \max_{x} \max_{f} L(x,f)$ when f is infinite dimensional

Consider a function $L: \mathbb{R}^n \times H \to \mathbb{R}$ with minimal assumptions on L. Here, H is some Hilbert space and we may assume it’s elements are smooth functions. So in one sense $L$ may ...
gabe's user avatar
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Which methods are best to analyse wave-packets?

I would like to find several methods to analyse wave-packets. My primary aim is to find a way to decompose wave-packets into "sub-wave packets" that, when subjected to some operation give ...
Luthier415Hz's user avatar
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3 votes
2 answers
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Functional derivative as "partial derivatives" of a functional

To show that $p$ is a stationary point of a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, one needs to guarantee that $f$'s directional derivative at $p$ is zero in every direction. To do so, it ...
ToposFan's user avatar
1 vote
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Property of a function equivalent to being the solution of a Variational method?

There are many problems in physics whose solution (a function on some space) can be obtained by a variational principle, namely, the solution $\phi$ makes some functional $S$ extremal: \begin{align} \...
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