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.
Questions tagged [functional-calculus]
205 questions
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Cauchy's integral formula for Cayley-Hamilton Theorem
I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise:
Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ matrix over $\mathbb C$ and $f(z) = \det(z-A)$ is…
Sam
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30
votes
1 answer
Why do zeta regularization and path integrals agree on functional determinants?
When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions.
The first definition is based on zeta function…
anon
- 155,259
21
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1 answer
A property of exponential of operators
Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by
$$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$
I am interested in this property:
If $x\in X$, such that the function $t\mapsto e^{tA}x$ is bounded on…
user165633
- 3,001
20
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3 answers
The matrix logarithm is well-defined - but how can we *algebraically* see that it is inverse to the exponential, as a finite polynomial?
This question is inspired by this which I saw earlier today. I started writing my answer, to share the insight that the matrix logarithm can be defined on matrices that do not have unit norm using an alternative technique.
Now, Sangchul has posted a…
FShrike
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10
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Searching two matrix A and B, such that exp(A+B)=exp(A)exp(B) but AB is not equal to BA.
We know that if two matrix $A$ and $B$ commutes then $\exp(A+B)=\exp(A)\exp(B)$. I am trying to find two matrix that does not commute but $\exp(A+B)=\exp(A)\exp(B)$ is true for them.
Can anybody give exact example. Thanks
Fin8ish
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10
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Exponential of the Laplacian operator as diffusion equation
Let $u$ be a function on a domain $\Omega$ with some fixed boundary condition.
I have recently seen a notation $e^{\tau \Delta}u$ as meaning the the time evolution of $u$ by diffusion for a time $\tau$. I'm curious where this notation comes from,…
rviertel
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Meromorphic symbol pseudodifferential operator
Context: For some function $f\in\mathcal{M}(\mathbb C)$ (meromorphic function on $\mathbb C$), I am interested in linear operators $T_f$ that act on functions of the form $g_a:x\mapsto \exp(ax)$ in the following way:
$$T_fg_a(x)=f(a)g_a(x).$$
Here…
Kolakoski54
- 1,862
7
votes
1 answer
Spectral measure and measurable functional calculus
I am learning about measurable functional calculus at the moment.
I have learned that given a operator $T \in B(H)$ that is self adjoint, there exists a measurable functional calculus $\overline{\Phi}:B_b(\sigma(T)) \rightarrow B(H)$, i.e. a map…
NTc5
- 1,161
7
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1 answer
Strong continuity of the Borel functional calculus
I have sometimes heard that the Borel functional calculus maps bounded pointwise convergent sequences of Borel functions to strongly convergent sequences of operators. I gather "sequence" is important here, due to the measure theory aspect, we…
Jeff
- 7,557
6
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1 answer
If $a$ and $b$ commute in a $C^*$-algebra and $a$ is normal, then $f(a)$ and $b$ commute for any continuous $f$
I'm trying to find a way to demonstrate the following:
Let $(A,*,\|\cdot\|)$ be a unital $C^*$-algebra. If $a,b\in A$ commute and $a\in A$ is normal (i.e. $a^*a=aa^*$), then for every continuous function $f:$Sp$(a)\to\mathbb{C}$, $f(a)$ and $b$…
user75206
6
votes
2 answers
Does linearity on all commutative subalgebras imply linearity on the whole algebra?
Let $\mathfrak A$ be a $C^\ast$-algebra and $\phi\colon\mathfrak A\to\mathbb C$ a continuous positive function whose restriction to any commutative sub-$C^\ast$-algebra is linear. Is $\phi$ linear on the whole of $\mathfrak A$?
If not, what if…
Olius
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Question about continuous functional calculus and its application
I recently started learning about the topic functional calculus. My problem is that I have no idea on how to use it for, say, solving problems, exercises etc.
Here is a short review of what I learned so far.
The idea behind functional calculus seems…
Philip
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6
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3 answers
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 space and the theory of spectral multiplicity" by Paul…
bayes2021
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6
votes
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If an element in a Banach algebra is anihilated by an analytic function then it must be algebraic.
Let $A$ be a Banach algebra, let $a\in A$ and suppose $f(a)=0$, where $f$ is an analytic function defined on an open set $U$ containing $\sigma(a)$. Prove that $a$ is algebraic in the sense that $p(a)=0$ for some polynomial $p$.
PS: I have just…
Ruy
- 20,073
6
votes
1 answer
Proving that the algebraic multiplicity of $\lambda$ is the rank of the projection $P.$
Let $A \in \mathbb M_n(\mathbb C)$ be a normal matrix and $\lambda$ be an eigenvalue of $A.$ Let $\phi : C(\sigma(A)) \longrightarrow \mathbb M_n (\mathbb C)$ be the continuous functional calculus associated to $A.$ Consider the projection $P : =…
ACB
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