**2**

votes

**1**answer

139 views

### Eigenfunctions of an infinite summation operator

I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*:
$f \rightarrow \sum_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...

**2**

votes

**2**answers

286 views

### Geometrical interpretation of a Schrödinger operator

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function
$$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$
such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...

**2**

votes

**1**answer

115 views

### Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra

As the comment of Andreas Thom indicated here, a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero ...

**3**

votes

**1**answer

119 views

### Simple $Z^{*}$ algebra

What is an example of a simple $C^{*}$ algebra which all elements are (two sided or equivalently one sided) zero divisor?

**2**

votes

**0**answers

70 views

### A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor.
Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...

**3**

votes

**1**answer

164 views

### A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
To what extent ...

**4**

votes

**0**answers

110 views

### The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now ...

**1**

vote

**0**answers

146 views

### Applications of composition operators on Sobolev spaces

I wold like to know some examples where composition operators on Sobolev spaces are useful. I'm in the following situation.
$L^1_p(D)$ - homogeneous Sobolev space, in other words space of locally ...

**1**

vote

**1**answer

96 views

### Limit-circle and limit-point at endpoints

I was wondering if the following holds:
If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...

**3**

votes

**2**answers

172 views

### Schrödinger operators on a sphere

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...

**2**

votes

**1**answer

79 views

### A Possible characterization of F.D or AF commutative $C^{*}$ algebras

By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...

**0**

votes

**0**answers

117 views

### Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq ...

**6**

votes

**1**answer

211 views

### Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...

**0**

votes

**0**answers

44 views

### Self-adjointness of the components of the magnetic derivative

On $L^{2}(\mathbb{R}^{n})$ define the operator $\Pi_{j} u := (-i\partial/\partial x_{j} - A_{j})u$, where $A_{j} \in L^{2}_{loc}(\mathbb{R}^{n})$ represents the $j$-th component of the magnetic ...

**11**

votes

**1**answer

373 views

### What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...

**2**

votes

**0**answers

100 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**1**

vote

**0**answers

69 views

### Semigroups on Banach Lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that
$$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$
Where $X_+$ denotes the positive ...

**1**

vote

**0**answers

105 views

### Norm of the operator acting on spinor bundle

Please forgive me if the question is too elementary, but however I was unable to manage by myself. The question comes from J.Varilly, H.Figueroa and J. Gracia-Bondia book "Elements of noncommutative ...

**6**

votes

**0**answers

84 views

### How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...

**1**

vote

**0**answers

86 views

### Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
...

**4**

votes

**2**answers

241 views

### Lecture notes on semi group theory for linear evolution equations

I am reading (or trying to read :)) One parameter semigroups for Linear Evolution equations by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture notes ...

**0**

votes

**1**answer

72 views

### Schur norm for self-adjoint operators

If $A$ is a $n \times n$ complex matrix then the Schur norm of $A$ is given by $$ || A||_S := \max_{||B||=1} ||A*B||,$$ where $||. ||$ is the operator norm and $*$ is the Hadamard (entry-wise) ...

**6**

votes

**2**answers

315 views

### Can the boundedness of $A^2$ imply the boundedness of $A$?

Given an operator $A \in \mathcal L(B)$, $B$ being a Banach space, I came across the following question: assume $\mathrm{dom}(A)=\mathrm{range}(A)$, $\mathrm{dom}(A)$ dense in $B$.
Under which ...

**4**

votes

**1**answer

171 views

### Left invertible operators of $B(X,Y)$

Suppose that $X$ and $Y$ are Banach spaces. Is $\{f\in B(X,Y):f\ \text{has a left inverse}\}$ an open subset of $B(X,Y)$?

**0**

votes

**1**answer

48 views

### Strongly convergent series of bounded self-adjoint operators

Let $A_n$ and $A$ be bounded self-adjoint operators in a Hilbert space, such that $A_n\to A$ strongly. Then it is well known that $(z-A_n)^{-1}\to(z-A)^{-1}$ strongly for each ...

**6**

votes

**3**answers

417 views

### Invertible operator with countable spectrum

Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set?
Motivation:
...

**2**

votes

**1**answer

118 views

### Invertible unbounded linear maps defined on a Hilbert space

It is well-known that, assuming the axiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?

**2**

votes

**2**answers

254 views

### A question on unbounded operators

Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:
Every densely defined operator $A:D(A)\to ...

**3**

votes

**1**answer

205 views

### Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...

**0**

votes

**1**answer

139 views

### Densely-defined unbounded operators with large support

Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense.
Let us consider ...

**2**

votes

**1**answer

186 views

### Commutator with Dirichlet Laplacian

Consider the Dirichlet Laplacian $\Delta$ on a compact Riemannian manifold (with boundary). Consider the operator $T = \sqrt{-\Delta}$. My question is: is there any Leibniz/product rule? Can we say, ...

**8**

votes

**1**answer

316 views

### How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...

**3**

votes

**1**answer

159 views

### Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times ...

**4**

votes

**1**answer

183 views

### Asymptotic behavior of Schrödinger operators

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

**2**

votes

**1**answer

216 views

### pick interpolation — why is it symmetric? $\left[\frac{1 - w_i \overline{w_j}}{1 - z_i \overline{z_j}} \right]_{i,j=1}^n \geq 0$ [closed]

I am reading notes on a complex interpolation problem:
Let $z_1, \dots, z_n \in \mathbb{D}$ and $w_1, \dots, w_n \in \mathbb{C}$. There exists (bounded holomorphic?) $f \in H^\infty(\mathbb{D})$ ...

**5**

votes

**1**answer

159 views

### Symplectic Koopmanism

Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and let $L_2(M,|\omega^n|)$ be the Hilbert space of complex-valued functions on $M$ that are square integrable with respect to the Liouville ...

**4**

votes

**0**answers

233 views

### Denseness of finite rank operators in $\mathcal{B}(X,Y)$

Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
...

**9**

votes

**2**answers

406 views

### Continuity of the product map

Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...

**2**

votes

**0**answers

64 views

### Necessity of coercivity assumption in Minty's theorem

Minty's Theorem states that a bounded, Continuous, monotone and coercive function on a Hilbert space is surjective.
A function $f:H\to H$ is called coercive if
$$ \lim_{\|x\|\to \infty} \frac{\langle ...

**4**

votes

**0**answers

177 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 ...

**1**

vote

**2**answers

214 views

### Continuous linear functionals in strong operator and $\sigma$-strong topologies

It was mentioned in the comments to http://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh that continuous linear functionals on ...

**1**

vote

**0**answers

81 views

### Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...

**8**

votes

**2**answers

670 views

### Mathematical equivalent to ladder operators?

A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...

**1**

vote

**1**answer

107 views

### Operators from $\ell_\infty$

It is well-known that non-weakly compact operators from $\ell_\infty$ into any Banach space act as isomorphisms on some subspace of $\ell_\infty$ isomorphic to $\ell_\infty$. I have a question in this ...

**2**

votes

**0**answers

177 views

### Predual of a von Neumann algebra in terms of trace class operators

For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ ...

**0**

votes

**0**answers

152 views

### Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?

$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...

**0**

votes

**0**answers

58 views

### A theorem of Lalesco on Trace-Class operators

There is this necessary and sufficient condition for nuclreaity of integral operators due to S.Chang and Lalesco that states : $$T_K \in\mathcal{S}_1\:\iff\:\exists K_1,K_2\in L^2([a,b]\times[a,b])\: ...

**0**

votes

**1**answer

110 views

### Solvability of an integral equation

Is it possible to find $f,g\in L^2[(0,1)\times(0,1)]$ such that $$\log|x-y|=\int_0^1f(x,t)g(t,y)dt\:\:\:\forall x,y\in(0,1).$$

**1**

vote

**0**answers

128 views

### If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research:
Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...

**1**

vote

**1**answer

240 views

### A property of groups of operators

Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...