# Tagged Questions

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### Inverse of operator is not continuous in Banach spaces

Let $X$ be a Banach space. If $A:X\to X$ is an invertible bounded operator (injective, surjective and continuous), then $A^{-1}$ is also bounded. Now can I have an example of an unbounded operator ...
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### decomposition of spectrum in case of a Banach algebra

Let A a complex Banach algebra containing a unit e. For x in A you can define the spectrum $\sigma(x)$ and the resolvent set of x. But, if you consider a complex Banach space X and a bounded linear ...
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### Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
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### Inverse of an operator on two functions

I have the following operator, defined for two twice-differentiable functions $f,g$: $X(f,g):=\frac{(g')^3+fg'f''+g'(f')^2-ff'g''}{g'f''-f'g''}$ This operator has the following property: A curve ...
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### Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
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### Why does T symmetric imply T* extends T?

This is a result I've seen stated a few times, but I can't seem to come up with a proof! Suppose $T$ is a densely defined linear operator with domain $D(T)\subset H$, where $H$ is a Hilbert space ...
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### Definition and analyticity of $T^z$ where $T$ is a positive operator

Let $H$ be a Hilbert space. Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of ...
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### Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
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### Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
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### The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
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### Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
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### Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi]$ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
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Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$
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This might be a possible duplicate - please let me know if there is already a proof in another thread. Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$: ...
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### Unbounded Operators: Notation?

For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$. Is there a notation for ...
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### The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
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### Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...