# Tagged Questions

Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

30 views

### How to do spectral decomposition?

I missed the last couple classes due to a family emergency and am trying to catch up with review questions. However, I can't seem to find an online source that teaches how to compute a spectral ...
29 views

### compactness in $\ell^2$

How can I show T is compact when T is defined as $$\text{T :}\,\ell^2 \to\ell^2\,\text{by Tx=y where} \,y_j=\alpha_jx_j\text{and}\,\alpha_j\to0\,\text{as}\,n\to\infty$$
33 views

### Intuitive understanding of quantum ergodicity of eigenfunctions

I recently heard a talk on differential geometry where the speaker was using a result called quantum ergodicity of eigenfunctions. I am trying to see if I am getting the gist of the result correctly. ...
26 views

### Spectral Theorem for compact Operators

I think about the spectral theorem for compact operators on a Banach Space. And I come to a question: Can the Theorem be generalized to any Normed Space or a bigger subclass of TVS
58 views

31 views

25 views

### Computing spectra in Banach algebras

In general, computing the spectrum of a specific element in a Banach algebra can be very difficult. What are some of the less obvious tricks that you've encountered?
29 views

### How to find spectrum of a convolution operator

Say $k$ be s.t. $\hat{k}$ is a bounded function on an LCA group $G$ and $Tf=f*k$. Then $T$ is bounded on $L^2(G)$. Is there anything I can say about $\sigma(T)$? (except the properties that follow ...
23 views

### Holomorphic Functional calculus & approximate point spectrum

Let $A\in B\mathbb{(H)}$ and $f\in Hol(A)$,then $f({\sigma}_{ap}(A))={\sigma}_{ap}(f(A))$. I only know how to prove $f({\sigma}_{ap}(A))\subset{\sigma}_{ap}(f(A))$,but have no idea on the other side. ...
99 views

### Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
66 views

23 views

Give an example of a Hilbert space $\mathcal{H}$ and an operator $A: \mathcal{H} \to \mathcal{H}$ satisfying $$\sigma \left( A \right) = \varnothing \neq \sigma \left( A^2 \right),$$ where $\sigma \... 0answers 46 views ### Finding the spectral representation of the Delta function given the Green function of an operator I'm working through the problems of a book on Sturm-Liouville problems. In a problem I found the Green function for the following SLP2 problem $$\frac{-d^2g}{dx^2}-\lambda g=\delta(x-\xi)$$ $$g(0,\... 0answers 50 views ### Complete orthonormal system of eigenfunctions for trace-class nonnegative operator on a Hilbert space In Da Prato/Zabczyk's book "Stochastic equations in inifinite dimension" I stumbled over the following paragraph: Let Q be a trace class nonnegative operator on a Hilbert space U. [...] Note that ... 0answers 57 views ### H self-adjoint with mass gap, P \ge 0,\Omega \in D(P), H + \lambda P self-adjoint \implies for \lambda small, H+ \lambda P has gap? Suppose H is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue 0 with gap 1 ( H\Omega = 0, \Vert \Omega\Vert = 1 ), P is a non-negative symmetric ... 0answers 17 views ### Abel transform of R_z(\Delta) The spectrum of the Laplace operator \Delta, as self-adjoint unbounded operator on L^2 is equal ]-\infty ,0]. The resolvent R_z is defined for z \in \mathbb C \setminus ]-\infty ,0] . Using ... 1answer 16 views ### rad(T)=||T|| for non-normal T It is well-known that for normal bounded operators T on a Hilbert space one has \mathrm{rad}(T)=\|T\| (where rad is the spectral radius). Are there any sufficient conditions under which a non-... 0answers 76 views ### Eigenvalues of a certain product of matrices with special structure Sorry for cross-posting from MO. Let d and c be positive integers and q = dc. Let G be a q-by-q positive semi-definite real matrix with eigenvalues all \le 1, and define the q-by-2q... 0answers 38 views ### operator ideals and their relationship to the geometry of Banach spaces, and other questions [closed] Albrecht Pietsch wrote an excellent book on operator ideals in 1978, which I use frequently. But, I was reading the preface to his book, and I do not understand it. He writes (in English translation)... 2answers 207 views ### Is \int |x\rangle \langle x|dx Mathematical? I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ... 1answer 96 views ### proving a quadratic form is closed I'm trying to show that, given a spectral measure d\mu_\psi(\lambda) for a self-adjont operator A, for the following quadratic form$$q_\lambda(\psi)=\int_{\mathbb R}\chi_{(-\infty,\lambda]}(\tau)... 0answers 41 views ### Proposed proof Decomposition theorem Let$\mathcal{A}$be a unital Banach algebra. I want to prove that if$a \in \mathcal{A}$and spectrum$\sigma(a) = \sigma_{1} \cup \sigma_{2}$where$\sigma_{1} \cap \sigma_{2} = \emptyset$,$\sigma_{...
I was given the following question in my linear algebra course. Let $A$ be a symmetric matrix, $c >0$, and $B=cA$, find the relationship between the spectral decompositions of $A$ and $B$. ...