# 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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### Find the norm of S

Given $C[0,2]$ with the the max-norm. Let $X= \{x\in C ([0,2]): x (1) =0\}$. We define $S:X\to\mathbb{R}$ as $S(x)=\int_{0}^{2}x(t)dt$ . Compute $||S||$. I have already found that $||S||\leq2$, but I ...
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### Riesz idempotent (projection) of singleton in $\ell^p$

Edit: After I posted my question, I realised that in this post the same question was asked (and answered). In the meantime useful comments were made under my current post. I've written an answer ...
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### Proof of continuous functional calculus via BLT

Background: I'm working on a proof of the spectral theorem as given by Halmos. As I've figured out, the first step, which is omitted in the proof, is to define a continuous functional calculus, i.e. ...
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### Range of positive operator

Let $E,F$ be Hilbert $A$-modules, and $T \in \mathcal{L}(E,F)$ with $\lVert T \rVert \leq 1$. Then, $1-T^*T$ has dense range iff $1-TT^*$ has dense range. This is Lemma 10.3 in Lance's book, but I don'...
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### Can one reduce the study of unbounded operators to the one of bounded operators?

So I am not very well-versed in functional analysis, but while studying a problem from theoretical solid-state physics I came across the following question: Suppose $\mathcal{H}$ is a complex Hilbert-...
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### Why is the Spectrum of an Operator Used as the Domain in Continuous Functional Calculus?

I'm currently working to grasp the concepts of (continuous) functional calculus, aiming to prove the spectral theorem for bounded self-adjoint operators as outlined in "Introduction to Hilbert ...
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### How to simultaneously remove a convolution and a multiplication

If you have an equation of the form $$a(x) \star f(x) + b(x)f(x)=0$$ Where $\star$ is convolution over the real line and $a,b$ are given functions where you want to solve for the set of possible $f$....
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### When is the image of the functional calculus of a self-adjoint operator also self-adjoint?

Let $A$ be a self-adjoint operator, either bounded or unbounded, and $f$ a Borel function. Using the functional calculus we may define $f(A)$ as a linear operator. Are there any known conditions on $f$...
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