# 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.

161 questions
22 views

### 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 ...
31 views

### 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 ...
37 views

98 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 $-\Delta$ ...
176 views

89 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 $\left( X, \mu \right)$...
48 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 $a \in A$ ...
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 ...
246 views

For a function $f(x)$, it is possible to write it as a taylor series centered around a point $x=a$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a){(x-a)}^{n}}{n!}=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}}{2}... 1answer 51 views ### If f \in C(\sigma(a)) and g \in C(\sigma(f(a))), proof that (g \circ f)(a) = g(f(a)) Let A be an unital C^\ast- algebra and let a \in A be normal. If f \in C(\sigma(a)) and g \in C(\sigma(f(a))), where \sigma(a) is the spectrum of a in A and \sigma(f(a)) is the ... 0answers 33 views ### Compute \frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau I need to compute:$$ \frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau, $$where t\in (0,T), A\in\mathbb R^{n\times n} and x\in\mathbb R^n. Using Leibniz differentiation under an integral sign, I ... 2answers 79 views ### Showing that the Holomorphic Functional Calculus preserves adjoints. Let T\in B(X) for some complex Banach space X. For any holomorphic f on \Omega\supset \sigma(T) I'd like to show that f(T^*)=f(T)^*, where f(T) is defined via the Holomorphic Functional ... 1answer 47 views ### Holomorphic functional calculus proving a property of fractional powers Consider the set S=\{re^{i\theta}:r>0,-\pi<\theta<\pi\}, i.e just \mathbb C without a branch cut. Let T\in B(X) for some Banach space X with \sigma(T)\subset S, and let \alpha,\... 1answer 73 views ### Is this a Functional Differential Equation? How to solve it? I ran into the equation below. I'm not familiar with functional derivatives so I'd appreciate if someone could give me an idea of how to solve it and/or a good reference I can use. I appreciate your ... 1answer 105 views ### Consider the operator T:L^2(0,1)\rightarrow L^2(0,1). Is it well defined, linear, bounded, compact? I'm studying Funtional Analysis and I'm doing some exercise, but I haven't got any idea how to start this. Consider the operator T:L^2(0,1)\rightarrow L^2(0,1) given by$$(Tu)(x):=\int_0^1(x^2+3x+1)...
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 ...