Questions tagged [functional-calculus]

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6
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1answer
153 views

An inverse to functional calculus

Given a Borel function $f:\mathbb{R}\rightarrow\mathbb{R}\cup\{\infty\}$, functional calculus allows to calculate $F(x)$ for any unbounded selfadjoint operator $x$ on a Hilbert space $\mathcal{H}$, ...
2
votes
4answers
190 views

EM-wave equation in matter from Lagrangian

Note I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success. Setup Let's suppose a ...
0
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0answers
41 views

Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $\ell^{\infty}(\mathbb{N})$ Unbounded Knapsack Problem $ \max_{x_1,...,x_n} \sum_{i=1}^n v_ix_i$ $\text{ subject to }$ $\sum_{...
3
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0answers
140 views

Gelfand "Calculus of Variation" 1.7 question on definition and purpose of variational derivative

In Gelfand Calculus of Variation, chapter 1.7, the variational derivative is defined as: $$\left.\frac{\partial J}{\partial y}\right|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\...
0
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0answers
129 views

A gap in the proof of uniqueness of functional calculus based on a spectral theorem

This question considers the proof of a fundamental theorem of functional calculus, given in the book Spectral Theory - Basic Concepts and Applications by David Borthwick (Theorem 5.9). Firstly we have ...
0
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0answers
53 views

"Trade-off" between bound on the function and on the spectrum for functional calculus in spectral theory

Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$. From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...
4
votes
1answer
160 views

Hölder continuity of functional calculus

Let $0<\beta<1$ and $ f \colon [0,1] \to [0,1]$ be $\beta$ Hölder continuous with constant $C$. Let $H$ be a Hilbert space and $A,B$ be self adjoint operators on $H$, such that $\sigma(A+B),\...
2
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0answers
84 views

Can all (inverse) trigonometric functions with periodic iterates be characterized?

I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
1
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1answer
196 views

Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let $H$ be ...
4
votes
0answers
81 views

Pseudodifferential Operators and Functional Calculus

I hope this is not too naive a question for MO. I've been taking a mathematical physics course, and was shown how operators like $\sqrt{1-\Delta}$ could be defined by taking multiplication operators ...
3
votes
0answers
55 views

A strange convergence for a semigroup of operators

I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows: Let $A,B$ ...
0
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2answers
66 views

The derivative of a $C_0$-semigroup with respect to a perturbation parameter

Let $H$ be a Hilbert space, and $A : H \to H$ be the (semi-bounded) generator of the $1$-parameter $C_0$-semigroup $[0, \infty) \ni t \mapsto \mathrm e ^{-t A}$. Let $B : H \to H$ be a bounded ...
1
vote
1answer
309 views

How to compute integral of a gaussian over a noncentered ball?

Let $\mathcal{B}(x,r)$ the ball of center $x \in \mathbb{R}^n$ and radius $r>0$ (so $\mathcal{B}(x,r) = \{y \in \mathbb{R}^n : \|y-x\| \leq r\}$, where all norms are $\ell^2$-norms). I would like ...
1
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1answer
76 views

Variational problem: how to minimise the second moment?

This is a neater version of a question I posted here, on which I'm also stuck. The problem: Say I have a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-...
2
votes
2answers
205 views

Hilbert Scale Inclusions

I'm looking at properties of the scale of Hilbert spaces $(X_s)_{s\in \mathbb{R}}$, which are constructed as follows. Starting with $A:D(A)\subset H\to H$, $A$ a densely defined, strictly positive ($...
2
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0answers
116 views

Linear independence of functions

Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
2
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0answers
63 views

Reference on iterated integrals against projection valued measures

I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
3
votes
1answer
118 views

The imaginary exponential of a tangent field on a manifold

If $M$ is a compact Riemannian manifold and $X$ is a tangent field, I am seeking to define the object $\exp {\mathrm i t X}$ for $t \in \mathbb R$, and I do not know how to do it. One option was to ...
1
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0answers
72 views

Condition for the integrability of a matrix function

Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\...
2
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0answers
159 views

Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
5
votes
1answer
446 views

Unbounded version of continuous functional calculus

For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
0
votes
1answer
248 views

Background on the functional equation $F(x+1)+F(x)=f(x)‎$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, ‎the following ‎difference ‎functional ‎equation ‎and ‎its ‎solutions ‎are ‎very ‎important: ‎$$‎\bigtriangleup ‎F(x):=F(x+1)-...
0
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1answer
131 views

Does Borel functional calculus commute with *-isomorphism?

I am confused with the underlined equation in the following picture. I know that a *-isomorphism commutes with continuous functional calculus since every continuous functions on the compact subset of ...
0
votes
1answer
183 views

For $B=\int \lambda d E_\lambda $ and $X$ commutes with every $E_\lambda $, why $BX$ is positive and self-adjoint?

Let $B$ be an unbounded closed operator on a Hilbert space $H$. If $B=\int \lambda d E_\lambda $ is positive self-adjoint and a positive bounded operator $X$ commutes with every $E_\lambda $, then why ...
0
votes
1answer
129 views

Variation in Einstein-Hilbert action [closed]

In this page there are calculations of variation of Einstein-Hilbert action. I see variations of terms like this: $\delta {R^{\rho }}_{{\sigma \mu \nu }}$ where the term is not a functional, and ...
3
votes
1answer
270 views

Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?

I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow. For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be ...
0
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1answer
246 views

Converse of Lax-Milgram theorem [closed]

Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V. Assume that, for any continuous linear functional on $l \in V’...
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0answers
105 views

Sufficient condition for gradient existence in Hilbert spaces

Let $\mathbb H$ a Hilbert space and $N:\mathbb H\to \mathbb H$ a continuous nonlinear mapping. In Fonda and Mawhin (Iterative and variational methods for the solvability of some semilinear equations ...
6
votes
3answers
394 views

Differential calculus of functions of self-adjoint operators

Let $H$ be a Hilbert space over $\mathbb{C}$. Fix a self-adjoint operator $A:D(A)\rightarrow H$ and a Borel function $f:\mathbb{R}\rightarrow\mathbb{C}$. The operator $f(A)$ is defined by the spectral ...
1
vote
1answer
167 views

Optimal joint coupling of all probability measures on a 3 point space

I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful. Consider a discrete probability space $\Omega = \{x, y, ...
4
votes
1answer
306 views

Reference Request: Calculus of Variations in Hilbert Space

I'm looking for a good reference to a book on calculus of variations in the setting of Banach Spaces. If it helps, I'm working with a particular functional acting on Fr\'{e}chet-differentiable ...
1
vote
0answers
120 views

infinite dimensional funtional ito calculus

I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
3
votes
1answer
261 views

Integrating the resolvent of a self-adjoint operator across a continuous part of the spectrum

Let $A$ be a closed self-adjoint operator on a Hilbert space $H$, possibly unbounded and hence defined on a dense domain $D(A) \subset H$. It is well known that integrating the resolvent $R_z = (z I - ...