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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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Spectrum of an operator defined by spectral integral

First of all I want to thank you for the help you provide on this website! Whenever I had a hard time understanding things in math I visited this website and (nearly) allways found a hint or a ...
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Product of two functions with compactly supported Fourier transforms

Question 1: Suppose $a,b\in C^\infty(\mathbb{R})\cap L^\infty(\mathbb{R})$. Is it possible that the pointwise product $ab$ equals the constant function $1$ and both $a$ and $b$ have compactly ...
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Interpreting Volterra Series Correctly

I was reading this and found on page 6 was a description of a generalization of Taylor series to linear functionals. Reproduced below as $$ F[\phi + \lambda] = \sum_{n=0}^{\infty} \left[\frac{1}{n!} ...
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Derivative of a functional and taylor series

In the book "quantum field theory for the gifted amateur", the authors used the derivation shown below:- I have trouble understanding the step in equation 1.16. Shouldn't the term $\frac{dg(f')}{df'} $...
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A book for questions on functions

I recently learnt functional analysis and I'm pretty comfortable at solving its questions but when it comes to more complex statements(though it may not be complex but for me they seem they are)I get ...
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Compact operator, functional calculus

Let $H$ be a Hilbert space, T be a compact operator and $f$ be a bounded function on $\sigma(T)$. Now I want to show that the operator $f(T)$ (in the sense of functional calculus) is compact if $dim(H)...
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Bounded function of compact normal operator on Hilbert space is normal

Let $H$ be a Hilbert space and consider a compact normal linear operator $A:H \to H$. Moreover, let $f$ be a bounded function on the spectrum $\sigma(A)$ of $A$ and consider the operator $f(A)$ in the ...
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Functional Derivative for Specific Question

Can you help me understanding how author got to equation 1.12, and what is phi(X)function. (https://i.stack.imgur.com/16LOQ.jpg) $$J[f] = \int [f(y)]^p \phi{(y)} d{...
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Inverting the Laplacian

I've had a hard time looking for literature on this, so here's my question: We take a look at the Laplacian $-\Delta$ as an unbounded operator on $\mathrm{L}^2(\mathbb{R}^3)$. We know that $-\Delta$ ...
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In the Physicists' definition of the path integral, does the result depend on the choice of partitions?

The standard definition of the path integral in Quantum Mechanics usually goes as follows: Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,...
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About velocity and acceleration as derivatives [duplicate]

Why is velocity the derivative of position with respect to time? Why is acceleration the derivative of velocity with respect to position? Is there any way to understand this logically, intuitively? Is ...
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Definite integrals : how do we approach in solving a problem

While practicing definite integrals I came across a question and now I am stuck Question: let f be a continous satisfying $f(x+y) = f(x) + f(y) + f(x)\cdot f(y)$ for all real $x$ and $y$ and $f'(0)=...
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Proving that $|x^*(x)| \leq \rho(x,Y)$

Exercise : Let $(X,\|\cdot\|)$ be a normed space, $Y$ a subspace of $X$ and $x^* \in X$ with $\|x^*\| \leq 1$ such that $x^*|_Y = 0$. Show that $\forall x \in X \setminus Y$, it is : $|x^*(x)| \leq ...
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Eigenfunction in functional calculus

Let $X$ be a complex Banach space, $A\in L(X)$ and $F$ be an analytic function in a neighborhood of $\sigma(A)$. Now I want to show that if $x\in X$ is an eigenfunction of $A$ corresponding to the ...
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Invertitbility of operator in functional calculus

Let $X$ be a complex banach space and $A$ be a bounded linear operator from $X$ to $X$. Further, let $F$ be an analytic function in a neighborhood of $\sigma(A)$ such that $1/F$ is an analytic ...
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Operator norm convergence in functional calculus

Let $X$ be a complex Banach space. Suppose that $A:X \to X$ is a bounded linear operator and that $(F_n)_{n \in \mathbb{N}}$ is a sequence of analytic functions in a fixed neighbourhood $D \subset \...
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Differentiability of a multi variable function.

My question is, like in the case of complex differentiation for a function to be differentiable at a point (x,y) the derivative must be same no matter which direction is chosen. So, if I have a ...
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Functional derivative normalization sensitive to normalization of test function?

Background: I am a physicist with decent background in mathematics. Reading the article on functional derivatives on wikipedia gives: The functional derivative $\frac{\delta F}{\delta \rho(x)}$ of ...
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Dimension of subspace annihilated by set of linear functional.

Let F be subfield of complex number We define n linear functional as $f_k(x_1,....x_n)=\sum^n_{i=1}(k-i)x_i$ I wanted to find dimension of subspace annihilated by $f_1,...f_k$ My attempt I had ...
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How to simplify the following functional derivative?

I have to simplify the functional derivative: $$ \frac{\delta}{\delta f_{k}(\boldsymbol{x}',t)}\delta(\boldsymbol{x}-\boldsymbol{\xi}(t))$$ where $\delta(\boldsymbol{x}-\boldsymbol{\xi}(t))$ is a ...
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Terminology / reference request: Calculus of variations with constraints on the family of curves

I am interested in variational problems of the following form: $$ \max J(y) \quad\text{such that}\quad y\in C, $$ where $J(y)=\int_0^b f(x,y,y')\,dx$ is a functional and $C$ is a specified family of ...
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Minimizing a functional.

I am doing some work on Kernel based machine learning where I encountered the following functional - $$\frac{1}{m}\sum_{i=0}^m(y_i-f(x_i))^2+\gamma\lVert f\rVert^2$$ Here $\gamma$ is a positive real ...
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Functional derivative of a functional that depends on antiderivative

It is well known how to calculate functional derivative if a functional depends of the function and it's derivatives (Euler-Lagrange rule): $\mathcal{L}=\int F(x,\dot{x},t)dt$. There is also straight-...
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Is functional calculus continuous on elements of the algebra.

Suppose $A$ is a C*-algebra, $a$ is a hermitian element of $A$. For each continous function $f:\mathbb{R}_+\to \mathbb{C}$, we say $f$ is continuous on $A$ if for every sequence $\{a_\lambda\}$ of ...
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Definitions of fractional Sobolev Spaces

In a paper I read that for a bounded domain $W^{s,p}(\Omega)=\{ u \in L^p(\Omega), (id-\Delta)^{s/2}u \in L^p(\Omega) \}$ and $s$ is not assumed to be an integer, $p \neq 2$ in general. If $p=2$ the ...
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Functional derivative in QFT

Introductory overview I have that $$iW_0[J] := -\frac{1}{2}\int d^4x d^4 y J(x)D_F(x-y)J(y)$$ and I'm trying to perform the calculation of a two-point function $G^{(2)}(x,y)$ from the fully ...
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QFT - Generating Functional

The problem statement, all variables and given/known data Hi I am looking at the attached question part c) Relevant equations below The attempt at a solution so if i take $\frac{\partial^{(n-1)}}...
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1answer
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The inner product on the space bounded linear operators over a Hilbert space

I am reading book "A short course on Spectral Theory", written by William Averson and I got some stucks. Lemma 2.4.4 (page 54) claimed that: Let $A$ be a normal operator on a Hilbert space $H$ and ...
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Functional derivative involving solutions of a non-analytically solvable ODE

I am not a mathematician, so I apologize in advance for any sloppiness. Suppose we have the following differential equation: $$\dot{x}(t) = \Phi (x(t),\gamma)$$ where $\gamma$ is some parameter and $\...
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Calculus of variations with delayed/shifted variable

How can the following optimization problem be solved? $$ \min_{y} \quad \int_{-\infty}^{\infty} x^2y(x)\mathrm{d}x,\\ \mathrm{s.t.} \quad y(x)\leq \alpha y(x-d),\\ \int_{-\infty}^{\infty} y(x)\mathrm{...
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Continuous functional calculus of multiplication operator in $L_2$

I would like to calculate the continuous calculus of the multiplication operator by an essentially bounded function $\varphi : X \rightarrow \mathbb{R}$ in $L_2 (X, \mu)$, where $\left( X, \mu \right)$...
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C*-algebra pure states and functional calculus

Let $A$ be a commutative unital C*-algebra, and let $\tau$ be a state on $A$, so it is a linear functional on $A$ with norm 1 such that it takes positive elements to positive elements. Let $a \in A$ ...
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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 ...
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“Taylor Series” analog for functionals?

For a function $f(x)$, it is possible to write it as a taylor series centered around a point $x=a$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a){(x-a)}^{n}}{n!}=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}}{2}...
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If $f \in C(\sigma(a))$ and $g \in C(\sigma(f(a)))$, proof that $(g \circ f)(a) = g(f(a))$

Let $A$ be an unital $C^\ast$- algebra and let $a \in A$ be normal. If $f \in C(\sigma(a))$ and $g \in C(\sigma(f(a)))$, where $\sigma(a)$ is the spectrum of $a$ in $A$ and $\sigma(f(a))$ is the ...
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Compute $\frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau$

I need to compute: $$ \frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau, $$ where $t\in (0,T)$, $A\in\mathbb R^{n\times n}$ and $x\in\mathbb R^n$. Using Leibniz differentiation under an integral sign, I ...
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Showing that the Holomorphic Functional Calculus preserves adjoints.

Let $T\in B(X)$ for some complex Banach space $X$. For any holomorphic $f$ on $\Omega\supset \sigma(T)$ I'd like to show that $f(T^*)=f(T)^*$, where $f(T)$ is defined via the Holomorphic Functional ...
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Holomorphic functional calculus proving a property of fractional powers

Consider the set $S=\{re^{i\theta}:r>0,-\pi<\theta<\pi\}$, i.e just $\mathbb C$ without a branch cut. Let $T\in B(X)$ for some Banach space $X$ with $\sigma(T)\subset S$, and let $\alpha,\...
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Is this a Functional Differential Equation? How to solve it?

I ran into the equation below. I'm not familiar with functional derivatives so I'd appreciate if someone could give me an idea of how to solve it and/or a good reference I can use. I appreciate your ...
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Consider the operator $T:L^2(0,1)\rightarrow L^2(0,1)$. Is it well defined, linear, bounded, compact?

I'm studying Funtional Analysis and I'm doing some exercise, but I haven't got any idea how to start this. Consider the operator $T:L^2(0,1)\rightarrow L^2(0,1)$ given by $$(Tu)(x):=\int_0^1(x^2+3x+1)...
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Proving that a family of exponential operators has a uniform bound, without semigroup theory.

Set $T\in B(X)$ for some arbitrary Banach space $X$. Define the family of operators $\{e^{tT}\}_{t\in\mathbb R}$ via the holomorphic functional calculus. I have been able to prove, using basic facts ...
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How do I prove that the following function is increasing in $t\geq 1$ for any $1\leq y \leq t$?

I am sorry if I am asking something too specific and not useful to the general public, but I am stuck at proving that the following function is increasing for all $t\geq 1$ at any given $y$ such that $...
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proving existence of a particular linear operator on the space of bounded functions

I am having trouble solving this task: Let $X:=C_b([0,\infty))$ be the space of bounded functions $f:[0,\infty)\rightarrow\mathbb{R}$ equipped with the norm $\|f\|=\sup_{x\in[0,\infty)}|f(x)|$. ...
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About vectors that have bounded support representation on the spectrum of a self-adjoint operator

Let $A$ be an unbounded operator densely defined on an Hilbert space $X$. Let $A^*$ be its adjoint. Furthermore, suppose that the commutator satisfies $[A,A^*]=1$. If $\psi$ is an $\lambda$-...
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Is it possible to evaluate this indefinite integral?

Evaluate the Indefinite Integral: $$\int \sqrt{x - \sin(x)}\,dx $$ My attempt: I tried using Hypergeometric Functions but it doesn't seem to be of any help....
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Can a function be increasing or decreasing at a point?

I was solving: Determine the intervals of increase and decrease for $f(x) = \frac {2x}{ ln x}$ and I stumbled upon the fact that f(x) is decreasing on (0,e] and increasing on [e, $\infty$). This would ...
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Solving PDEs using the Ritz method on variational calculus problem (Student questions)

I'm reading the book "Conduction Heat Transfer" by Vedat S. Arpaci. I'm currently at chapter 8 (I didn't read the rest of the book, though), which talks about the Variational Formulation - Solution by ...
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Do we have $|BA|=B|A|$?

Let $B$ be an unbounded positive operator and $A$ be a bounded operator on a Hilbert space. Moreover, every spectral projection of $B$ commutes with $A$. Do we have $|BA|=B|A|$?
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$(AB)^p =A^p B^p$?

Let $A,B$ be two positive bounded linear operators on a Hilbert space. If $A$ commutes with $B$, do we have $(AB)^p =A^p B^p$ for any $p>0$? Or more general, $f(AB) = f(A)f(B)$ for any Borel ...
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400 views

Functional derivative of generating function QFT

I am reading Srednicki book for QFT. In page 69 they realize a functional derivative of the following function: $$ Z[J]= \exp\left(\frac{i}{2}\int d^4xd^4x'J(x)\Delta(x-x')J(x') \right) $$ with $$\...