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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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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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34 views

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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1answer
29 views

Given a linear operator $T$ and a linear functional $\phi_n(x)=(T(x))(n)$, show that $T$ is continuous iff $\phi \in X*$

Given $X$ a Banach space, and $T:X \rightarrow l_p$ a linear operator, with $1 \leq p \leq \infty$, for all $n \in \mathbb{N}$ consider the linear functional $\phi_n:X \rightarrow \mathbb{K}$, defined ...
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2answers
40 views

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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3answers
71 views

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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1answer
24 views

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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1answer
83 views

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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1answer
33 views

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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70 views

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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1answer
15 views

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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2answers
45 views

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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1answer
31 views

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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1answer
33 views

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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2answers
33 views

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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1answer
30 views

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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0answers
32 views

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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0answers
31 views

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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1answer
19 views

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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1answer
121 views

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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2answers
78 views

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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1answer
50 views

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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1answer
60 views

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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1answer
92 views

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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1answer
29 views

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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0answers
36 views

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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0answers
36 views

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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1answer
77 views

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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1answer
43 views

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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79 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 ...
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1answer
184 views

“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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1answer
43 views

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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0answers
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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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2answers
66 views

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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1answer
41 views

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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1answer
73 views

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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1answer
89 views

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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1answer
39 views

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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1answer
38 views

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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1answer
29 views

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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1answer
42 views

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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0answers
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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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5answers
558 views

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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1answer
469 views

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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0answers
41 views

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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1answer
54 views

$(AB)^p =A^p B^p$?

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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1answer
178 views

Borel Functional Calculus of Commuting Operators

let $H$ be a Hilbert space, $T_1,T_2 \in L(H)$ two self-adjoint operators such that $T_1T_2=T_2T_1 \in L(H)$, i.e. they commute and their product is continuous. $E_1,E_2$ are their assigned spectral ...
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2answers
59 views

Can't determine even function [closed]

Given $f(x)= \cos\left(\log( x+\sqrt{1+x^2}\right))$ As to check if even or not, we usually use $f(-x)=f(x)$ But here if we do that the even nor odd However when graph is plotted we get even. ...
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2answers
94 views

Searching for a proof that in a normed functional space of $C^0[0,1]$ with sup norm, that norm is nowhere differentiable.

Having a normed linear space $S=C^0[0,1]$ of continuous functions $f:[0,1] \rightarrow \Bbb R%$, with sup norm: $\|f\|=\sup_{\space x \in [0,1]}|f(x)|$, prove that $F(f)=\|f\|$ is nowhere ...
2
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2answers
45 views

Proving $(\lambda^{-1}-A^{-1})^{-1}=\lambda-\lambda^2(\lambda-A)^{-1}$

Let $X$ be a Banach space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, let $A\subset X\oplus X$ be a linear relation and let $\lambda\in\mathbb{K}\setminus\{0\}$ I want to prove that $(\lambda^{-1}-...
2
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1answer
44 views

Is $T(T^2)^{-1}$ a left-inverse for $T$?

Let $H$ be a Hilbert space and $T:\text{dom}(T)\rightarrow H$ be a densely defined, closed self-adjoint operator. Suppose that $T^2: \text{dom}(T^2)\rightarrow H$ has a bounded inverse $(T^2)^{-1}:H\...