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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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How to calculate the functional derivative of this composite function?

I think I am not unfamiliar with the functional derivative, while recently I encounter a paper which gives the functional derivative expression like this $$W=\ln[\sum_{N=0}^{\infty}\frac{1}{N!h^{3N}}e^...
Xeh Deng's user avatar
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What do people call an equation that describes a relationship between a functional and one or more of its functional derivatives?

"Functional differential equation" seems to already be used for something different (https://en.wikipedia.org/wiki/Functional_differential_equation). I'm talking about something like the ...
William Wright's user avatar
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Integration of composite function

I am confused about the integration of a nested function / composite function. Let's say the function is defined as follows \begin{equation} I(x) = f(g(x)) \end{equation} Obviously, when taking the ...
Dennis Marx's user avatar
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Understanding the integral of a bounded operator on a Hilbert Space

In mathematical physics I've encountered the holomorphic functional calculus, which I've seen the defined as: Let $f$ be holomorphic in a neighbourhood of the spectrum of $T$, with $T$ being a bounded ...
AJE's user avatar
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2 answers
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Domain of the inverse of a compact operator on Hilbert Space [closed]

i have the following problem: Let $A$ be an integral operator defined on $L^2([-1, 1])$ by: $ (Af)(x) = \int_{-1}^1 |x - y| f(y) \, dy.$ Consider the operator $T$ as the inverse of $A$, that is, $T = ...
Pietro Sileci's user avatar
6 votes
1 answer
118 views

Question about continuous functional calculus and its application

I recently started learning about the topic functional calculus. My problem is that I have no idea on how to use it for, say, solving problems, exercises etc. Here is a short review of what I learned ...
Philip's user avatar
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About differentiation of supernumber-valued functionals

I'm (trying) to read deWitt's A Global Approach to Quantum Field Theory and trying to get around the first definitions. The central object is a differentiable functional $F: \Phi \rightarrow \Lambda_{\...
Lourenco Entrudo's user avatar
4 votes
1 answer
117 views

Isomorphism in Banach Spaces

Let $E$ and $F$ be Banach spaces. Let $T: E \rightarrow F$ be an isomorphism (i.e., a continuous vector space isomorphism with a continuous inverse). Let $J_E$ and $J_F$ be the canonical injections of ...
Jason Jacob's user avatar
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The generator of analytic semigroup

Definition: Let $0<\omega<\frac{\pi}{2}$. A family of bounded operators $\{S(z):z\in S^0_{\omega_{+}}\}$ on $X$ is called a holomorphic semigroup if $S(z)S(\omega)=S(z+\omega) $ for all $z,\...
accretive's user avatar
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Multiplication of two unbounded operators and functional calculus

Let $A$ be a positive, self-adjoint unbounded operator defined on a Hilbert space $H$. Let $f,g: [0, \infty]\to \mathbb{R}$ be Borel measurable functions that are bounded on compact subsets. We can ...
Andromeda's user avatar
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How to prove the question about dual pair of Banach space?

Definition: Two Banach spaces $X$ and $Y$ form a dual pair $\langle X,Y\rangle$ if there is a bilinear or a sesquilinear form $\langle \cdot ,\cdot \rangle$ on $X\times Y$ which satisfies the ...
accretive's user avatar
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Pederson's Analysis now Proposition 4.5.10, Why is this operator equal to $0$?

Here is an image of the proposition: An highlighted below is the part of the proof I'm having trouble with. Why is the equation true for every n? $f_n(T) = \int_{\sigma(T)} f_n \: dE$, but I don't ...
Isochron's user avatar
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On functions of a self-adjoint operator of the form $U^{-1} A U$

I found in Reed Simon that, since $\mathcal{F}$ (Fourier Transform) is unitary in $L^2(\mathbb{R}^n)$ and the self-adjoint operator (in a suitable domain) $-\Delta = H_{0}$ can be expressed as $H_{0} =...
Alessandro Tassoni's user avatar
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Function equation separability-like criterion

I have a very specific problem that I'm not even sure how to approach. I have a function $G(t,s,W,W')$, such that $G(s,t,W,W') = -G(t,s,W,W')$, where $t$ and $s$ are real variables while $W$ is a real ...
peep's user avatar
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Proposition 4.4.12 Pederson Analysis Now, Why does this operator commute?

Here is the proposition: The part I'm not understanding is (as usual with most of Pederson's proofs) the last implication. The last equality just seems to imply that $S$ commutes with the power ...
Isochron's user avatar
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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
1 vote
3 answers
178 views

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
2 votes
0 answers
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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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1 answer
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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
1 vote
0 answers
65 views

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{...
LV2's user avatar
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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 ...
Linda's user avatar
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1 answer
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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 ...
Sha Vuklia's user avatar
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1 vote
1 answer
149 views

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. ...
bayes2021's user avatar
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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'...
mathable's user avatar
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3 votes
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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-...
Sylv_'s user avatar
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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 ...
bayes2021's user avatar
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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 $f$....
Sidharth Ghoshal's user avatar
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0 answers
16 views

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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0 answers
36 views

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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0 answers
69 views

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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2 votes
1 answer
65 views

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 $f$...
CBBAM's user avatar
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4 votes
0 answers
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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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1 vote
1 answer
129 views

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 $A_n \rightarrow A$ ...
CBBAM's user avatar
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0 answers
106 views

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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0 answers
34 views

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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0 answers
58 views

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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64 views

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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0 answers
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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 ...
CBBAM's user avatar
  • 6,255
0 votes
1 answer
87 views

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 $$\...
Lumos's user avatar
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1 vote
1 answer
63 views

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}\...
Lumos's user avatar
  • 121
4 votes
0 answers
228 views

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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2 votes
0 answers
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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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2 votes
0 answers
63 views

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
  • 1,020
0 votes
2 answers
65 views

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 ...
Superunknown's user avatar
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4 votes
0 answers
119 views

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
  • 81
2 votes
1 answer
68 views

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
3 votes
0 answers
49 views

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
  • 1,020
2 votes
0 answers
65 views

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
  • 6,255
-2 votes
1 answer
33 views

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
1 vote
0 answers
52 views

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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