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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1answer
41 views

Continuous functional calculus (Composition) [duplicate]

Let $A$ be $C^*$-algebra, $ x\in A$ be normal, and $f\in C(sp(x))$. Show that $(g\circ f)(x)=g(f(x))$ for all $g\in C(f(sp(x)))$. My attempt: I think it suffices to show that $C(f(sp(x))) \subseteq C(...
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Spectral decomposition and family of projections

I have been solving this problem, it is about spectral decomposition and and Borel functional calculus. However, I am new to the the theory. I am confused with the notation in my notes and in the ...
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1answer
35 views

Norm of the difference of two positive elements in a $C^\star$ algebra

Let $\mathcal{A}$ be a $C^\star$ algebra (assume it to be unital, if necessary), and $a,b\in \mathcal{A}_{+}$, that is, $a,b$ are positive elements. It is a well known fact that $\left\Vert{a-b}\right\...
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$U$ unitary: $\mathbb{T}\ne\sigma(U)$. Prove $\forall\varepsilon>0$ there exists a polynomial $p(z)$ such that $\|U^{-1}-p(U)\|<\varepsilon.$

Let $U$ be a unitary operator: $\mathbb{T}=\{\lambda:|\lambda|=1\}\setminus\sigma(U)\ne\varnothing$ (the spectrum does not cover the whole circle). Prove that $\forall\varepsilon>0$ there exists a ...
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Finding the minimizer of $I[f]=\int_1^2 (x^2 f'(x)^2+2f(x)^2)dx$

My Question is, if my solution is right. I am trying to minimize $$ I[f]=\int_1^2 (x^2 f'(x)^2+2f(x)^2)\,\mathrm dx $$ with the conditions $f(1)=0$, $f(2)=1$ Assuming $f$ is the minimizer and $\...
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1answer
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Why is functional calculus called functional calculus? What does functional mean in functional calculus?

Given a unital Banach algebra $B$, functional calculus allows one to define $f(a)\in B$ for continuous function $f$ and $a\in B$. A functional is a mapping from a vector space to its scalar space. But ...
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Spectrum of isometry not in circle

I have a reasoning that is wrong but I don't understand why. Suppose we have a unital $C^*$-algebra $A$ and $v \in A$ is a proper isometry, so $v^*v = 1$ but $v v^* \neq 1$. Take the function $f: \...
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Differential of fraction of function involve basis spline

Suppose, we want to find $\dfrac{\partial X_i(t)}{\partial c}$ where $X_i(t)=\dfrac{exp[Z_i(t)}{\int_a^dexp[z_i(s)]ds}$, $i=1,...,n$ and $Z(t)=b(t)^Tc + b(t)^TAU.$ Here $b(t)=[b_1(t),...b_p(t)]$ be a ...
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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: $\frac{\partial J}{\partial y}|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\Delta\sigma}$,...
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Matrix differentials to estimate the critical value from an equation that involves double summation and integration

The equation I have, $l(c)=-\sum_{i=1}^n\sum_{j=1}^{m_i}\dfrac{||y_{ij}-x_i(t_{ij})||^2}{2\sigma_e^2}$ where $x(t)=\dfrac{exp[b(t)^Tc+b(t)^TAu]}{\int_a^vexp[b(s)^Tc+b(s)^TAu]ds}$ Theoretically s or t ...
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Definition and notation of the functional derivative

In my lecture notes I came across the following definition of the functional (or variational) derivative using Dirac's delta: $$ \frac{\delta F\left[y\right]}{\delta y\left(x'\right)}=\left.\frac{\...
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Finding the minimizer of the variational problem $J[y]= \int_{0}^{2} \left( 1-y'^2 \right)^2dx $ with $y(0)=0$ and $y(2)=0$.

This is a question from a math contest. Find the minimizer of $J[y]= \int_{0}^{2} \left( 1-y'^2 \right)^2dx$ with the conditions: $y(0)=0$ and $y(2)=0$. Now as $F$ is $\geq 0$, so to minimize we let $...
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Polar Form of a Matrix (functional calculus )

Let $T= \begin{pmatrix} 1 & 1 \\ 0 & 0 \\ \end{pmatrix} $. Find the polar form of $T= W|T|$, where $|T|= \sqrt{T^*T}$, and $W$ is unitary. I was able to find $|T|= \begin{...
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Correspondence between divergent integrals and analythic functionals

The Dirac delta function can be defined the following way: $$\delta(x) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{itx}\, dt={\displaystyle {\begin{cases}\frac1{2\pi}\int_{-\infty}^\infty dt,&{\text{...
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Notational confusion in derivation of Euler-Lagrange equations

I'm reading Bishop's "Pattern Recognition and Machine Learning" section on the Calculus of Variations (Appendix D) and he defines the functional derivative of $\frac{\delta F}{\delta y(x)}$ ...
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Trying to derive the path integral from first principles (step 1)

I was recently told in the physics forum (https://physics.stackexchange.com/questions/616186/deriving-the-path-integral-from-the-time-slice-approach-for-a-general-hamiltonia) that it is not possible ...
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1answer
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functional calculus on C*-algebra and majoration of $||f(A+B)-f(A)||$

In introductory books on C*-algebra, functionnal calculus is quicly presented as it is a powerful tool in the field. It is for example prooved that for an operator $a$ and a function $f$ continous on ...
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Functional Calculus F(T)

I have to do the next problem but I'm a little lost. Let $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 & 0 \\ ...
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Understanding the spectral measure of unbounded self-adjoint operator

Let $\mathcal{H}$ be a Hilbert space and $$\Delta: D(\Delta) \rightarrow \mathcal{H}$$ a (unbounded) self-adjoint operator, densely defined on $D(\Delta) \subset \mathcal{H}$. Denote by $\mu^{\Delta}$ ...
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Chain rule for functional derivates with the same variables

I am following an online course where we should sometimes compute functional derivatives. The scope of the course is density functional theory, so we have the density $n(r)$ and we have functionals of ...
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Why can we use eigendecomposition in a funciton in this way

On this Wikipedia page, it says that $$f(A) = Qf(\Lambda)Q^{-1}$$ where $\Lambda$ is a diagonal matrix with the eigenvalues of $A$ on the diagonal, and $Q$ is the eigenvector matrix. In other words, ...
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Banach Algebra exponentials

Let $\mathcal{A}$ be a unital Banach Algebra. Suppose $a,b \in \mathcal{A}$. show that if $a$ and $b$ commute, i.e. ab=ba, then $\exp(a) \exp(b)= \exp(a+b)$. Give a counterexample to show that $\exp(...
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Transforming Functional Into Double Integral

I have a functional which I would like to minimise. It is of the form $$ F[x] = \alpha - \int_{0}^{1} \left(\int_{0}^{t} g(x(\bar{t})) \, d\bar{t} \right)^{1/3} f(\,x(t), \dot{x}(t)\,) \, dt \,\, , \...
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Convexity in functional space for GANs

In the paper: Generative Adversarial Nets, on Page 5 Proposition 2 Proof, the authors claim that equation (3) is convex in p_g. I am trying to understand how are the authors arriving at that ...
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1answer
43 views

Spectral Theory for “Unrepresented” C*-algebras

If you Google something like "Borel functional calculus" or "spectral decomposition" you get plenty of results for operators on Hilbert spaces. What about an (unrepresented) $\...
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(Relatively) Boundedness of self adjoint operator $A^q$.

Suppose $A$ is a self-adjoint operator on the Hilbertspace $\mathcal{H}$. Furthermore let $p,q \in \mathbb{R}$ with $0 \leq q < p$. I want to show that $A^q$ is relatively bounded w.r.t $A^p$ with $...
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The calculation of Gateaux derivative

How to calculate the Gateaux derivative of the following functional especially when $p$ is not a integer where $\Omega \subset \mathbb{R}^n$ is an open set ? $I(u)=\int_{\Omega} |u|^p dx$ Besides, how ...
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1answer
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Question about convergence of functions of an operator

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space. Assume that $f_n:\mathbb{R}\to\mathbb{C}$ is a sequence of continuous bounded functions such that $f_n \to f$ point wise. Then $...
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Functional derivative for compostion of functional

Is it possible to obtain the functional derivative of the following functional? $Z[q]=\int_0^T\int_0^t\int_0^t q(t'')K(t''-t')q(t')dt''dt'dt.$ I tried to apply the usual functional derivative ...
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1answer
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how is a first variation of a functional calculated?

given the functional: $$ F(\phi)=\frac{1}{2} \langle L\phi,\phi \rangle - \frac{1}{2} \langle \phi,f \rangle - \frac{1}{2} \langle f,\phi \rangle $$ where the differential equation with operator $L$ ...
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Can it be proved using something like functional calculus?

I have a question about a part of some proof. Let $\lambda$ be any number on the unit circle in the complex plane. Then, we know that there exists a real bounded Borel function $g(\lambda)$ such that $...
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Functional derivative of contour integral

I have the following contour integral $$F[\beta,\gamma] = \oint_\Gamma \frac{dz}{2\pi i} \beta(z)\gamma(z)$$ along some contour $\Gamma$ that encloses a pole of $\beta(z)$. I want to interpret $F$ as ...
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Does a functional have to be fed with the whole graph of a function?

Definition of functional goes like a map from vector space to field. If I consider vector space to be function space with single-variable-functions and field to be numbers then is it possible that ...
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1answer
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Functional derivative of a complex fourier sum

I'm trying to parse a derivation of a lagrange-multiplier fourier technique, and can't quite grasp an intermediate step. The author sets up $$ e_x(z,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty}...
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Chain rule for functional derivatives

I'm studying the functional derivatives on Mahler's book "Statistical Multisource-Multitarget Information Fusion". I have some difficulties in understand a chain rule for functional ...
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1answer
171 views

Functional derivative of the square of a functional

How should I compute the functional derivative of the following functional: $$ F(p(x)) = \left[\int\cos(x)p(x)dx - \int\cos(y)q(y)dy\right]^2 $$ I know that the functional derivative $\frac{\delta F}{\...
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1answer
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Derivative of a Functional. The Chain Rule? Whats going on here.

Hi suppose I have a functional $I_{x,y}$ acting on continuous paths $$\{\xi~:~\xi:[0,1]\to \mathbb{R}^n,~\xi(0)=x,~\xi(1)=y\}$$ . In my case $I_{x,y}[\xi]=\int_0^1 L(\xi(t),\dot{\xi}(t))dt$ where $L(\...
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Calculate the Euler-Lagrange for a functional with two nested integrals?

I've been reading papers about a fairly unknown topic in quantum mechanics called the quantum backflow effect. And in many of the papers they find an eigen value problem corresponding to the maximal ...
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1answer
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How to prove $f(A^*) =f(A)^*$

I'm studying Conway's functional calculus. If $\mathcal{X}$ is banach space and $A \in \mathcal{B}(\mathcal{X})$, and $f \in \mathtt{Hol}(A)$ , show that $f(A)^* = f(A^*)$ Firstly, I tried it for ...
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Functional Derivative with Discrete Variable

Problem $$\text{Find}\quad\frac{\delta F_k}{\delta G} \quad \text{given} \quad F_k=\left(\sum_{r=0}^{N-1} e^{ikr}\int_{-\infty}^{\infty} dt \ e^{i\omega t} G(r,t)\right)^{-1}$$ noting that $k$ and $r$ ...
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1answer
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An inequality of positive maps of roots of positive operators on C*-algebras

This is a pretty long question that I have been struggling with for a while, any tips or suggestions would be awesome! Let $A, B$ be two C*-algebras, the later algebra is represented on a complex ...
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1answer
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Understanding $z(a,b) \in[0,1]$ where $a=|x-y|$, $b=|y|$ and $x,y\in[-2,2]$

I previously asked a poorly written question that used incorrect terminology. A user @epiliam was very patient and worked me through my logic to understand what I was trying to ask. I had a question ...
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1answer
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The Tomita-Takesaki theory for von Neumann algebras with a separating and cyclic vector by Alfons van Daele.

I am learning the theory of standard von Neumann algebras by following Chapter $10$ from the book 'Lectures on von Neumann algebras' by Strătilă and Zsidó. But I got stuck with an argument provided in ...
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1answer
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Proof that $\frac{2x}{1+x}\leq f(x) \leq \frac{1+x}{2}$ for standard operator monotone function $f$

I am reading a text on operator monotones, defined as Definition 1 (Operator Monotone) A function $f:I\to\mathbb{R}$ defined on an interval $I \subset \mathbb{R}$ is said to be operator monotone if $$...
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1answer
55 views

Is there a name for this matrix transformation?

Let $A$ be a diagonalizable matrix over the field $F$ and $f:F\rightarrow F$. Then we can define the following matrix: $f(A) = P^{-1}f(D)P$ where $D$ is diagonal and $f(D)$ is the diagonal matrix ...
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Maximize functional with dot product in it

I have to find a function that maximizes some functional. This function can be divided into multiple functionals - as a first step, I wanted to find a function that maximizes only one of these (for ...
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How to do Taylor expansion to $g(\gamma_1(x),\gamma_2(x))=\frac{\gamma_1(x)}{\gamma_2(x)}$ around $(\gamma_{10}(x),\gamma_{20}(x))$?

Suppose $\gamma_1(x),\gamma_2(x)$ are both bounded and nonzero smooth real functions on $[a,b]\subset R$, and we define functional $g(\gamma_1(x),\gamma_2(x))=\frac{\gamma_1(x)}{\gamma_2(x)}$, how to ...
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How can we find the maximum?

Suppose that $y=y(a(\lambda),\lambda,x)$ is a decreasing and twice continuously differentiable functional with respect to $x$ and $a(\lambda)$ is linear mapping from a space $\Lambda$ to $(0,+\infty)$....
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1answer
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Confusion about definition (in ΨDO theory) of Sobolev spaces on open sets in Euclidean space

I'm reading Pseudodifferential Operators by M. E. Taylor, where the author talks about $H^s(\Omega)$ for $s\in\mathbb{R}$ and $\Omega\subset\mathbb{R}^n$ an open set (for example, in the statement of ...
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1answer
88 views

Exponential of commuting operators with holomorphic calculus

Let assume, for simplicity, a finite dimensional Banach space, and two commuting linear operators $A$ and $B$. By defining the exponential of some operator $T$ as $$ \exp(T):=\frac{1}{2\pi {\rm i}}\...

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