# Tagged Questions

**4**

votes

**1**answer

157 views

### Asymptotic behavior of Schrödinger operators

I am currently dealing with $1$ or at most $2$-dimensional SchrÃ¶dinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta ...

**4**

votes

**0**answers

148 views

### Adjoint of sum of two operators. Kato-Rellich

Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...

**0**

votes

**0**answers

73 views

### Is the exponential Mathieu operator trace-class?

Let $H \psi(x) = -\frac{d^2}{dx^2} \psi(x) - \alpha \cos(x) \psi(x)$ on $[0,2\pi]$
be the Mathieu operator ( according to Mathieu's ODE). My question is: Do we know whether $U(t):=e^{-tH}$ for some ...

**16**

votes

**4**answers

690 views

### Who first used the multiplication operator version of spectral theory

This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...

**0**

votes

**1**answer

47 views

### On a characterization of some subsets

Let $X$ be a Banach space. $A$ is a linear closed and densely defined operator and $S$ is a bounded invertible operator. What' s the relation between $\sigma_{e,S}(\lambda S - A)$ and ...

**0**

votes

**2**answers

276 views

### Spectral decomposition of compact operators

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...

**0**

votes

**1**answer

129 views

### Showing there is a unique spectral measure

All the books I have seen have proved that, for a normal bounded operator $T$, there is a unique spectral measure $E$ such that $\int_{\sigma(T)}^{}\lambda\,dE=T$ by first proving in it for a general ...

**5**

votes

**1**answer

133 views

### Is irreducibility sufficient for uniqueness of invariant distribution for a Feller Semigroup?

Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1 $ for every $t\geq 0$.
(This is also called a Feller Semigroup.)
...

**1**

vote

**0**answers

76 views

### “Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs).
Then $||AB||_{op} \leq ...

**0**

votes

**0**answers

68 views

### Spectrum of an operator from Transpose sum

I was wondering if there is anything we can know about the spectrum of an operator $A$ if we know that $M = A + A^{T}$ is a positive operator?

**0**

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**0**answers

94 views

### Perturbation of spectrum and eigenspaces

Let $A \in \mathbb{C}^{n \times n}$ be an $n \times n$ matrix. Consider the rank-$1$ perturbation $A'$ of $A$ given by replacing a column $v$ of $A$ by $\alpha \cdot v$, where $\alpha \in [0, 1)$. Can ...

**3**

votes

**2**answers

191 views

### A version of the spectral theorem for group actions

Suppose $G$ is a sufficiently nice (maybe locally compact and abelian) group which acts on the separable Hilbert space $\mathcal{H}$ by unitary transformations. Is there a generalization of the ...

**2**

votes

**0**answers

195 views

### Versions of the spectral theorem

Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:
($*$) ...

**4**

votes

**1**answer

156 views

### A.C. spectrum of the non additive perturbation BAB of a self-adjoint operator A where B is strictly positive

If have the following problem:
Let $A : \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on some Hilbert space $\mathcal{H}$. Let $B: \mathcal{H} \to \mathcal{H}$ be a bounded, ...

**6**

votes

**1**answer

238 views

### A doubt about the parts of the spectrum of tensor products

Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, ...

**1**

vote

**1**answer

244 views

### A is a nonnegative matrix; the only principal submatrix having spectral radius above 1 is A itself

Let $\rho(M)$ denote the spectral radius (modulus of the largest eigenvalue) of a square matrix $M$.
I am looking for a characterization or anything else interesting about the set of matrices $A$ ...

**5**

votes

**3**answers

566 views

### Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...