**0**

votes

**0**answers

55 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

103 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

102 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

233 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, ...

**16**

votes

**4**answers

736 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 ...

**-1**

votes

**1**answer

67 views

### On a characterization of some subsets [closed]

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 ...

**2**

votes

**1**answer

142 views

### Nuclear vs Integral operators on Hilbert spaces

Consider the Hilbert space $L_2=L_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L_2$ there exists a function $K\in L_1(L_2)$ such that
$$Tf = \int\limits_0^1 K(s) f(s) ...

**3**

votes

**1**answer

77 views

### Moments of the position operator and wavepacket spreading

I've noticed that when papers in mathematical physics concern themselves with the rate at which a wavepacket spreads, they almost always try to bound the moments of the position operator (the operator ...

**9**

votes

**1**answer

308 views

### A version of von Neumann inequality

Assume that $X,Y,Z$ are three commuting operators acting in a Hilbert space $H$. Assume also that they satisfy following properties:
1) $\|Z\| \le 1$, i.e. $Z$ is a contraction;
2) For any complex ...

**-4**

votes

**1**answer

271 views

### I need following books (soft copies) [closed]

I know this is not the place to ask for such help, but I cant find these books in my country and not even on line and the shipping is very expensive. If someone out there have any of these books (soft ...

**3**

votes

**2**answers

175 views

### Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some ...

**3**

votes

**0**answers

94 views

### Noncommutative Poincare inequalities

This is a question on how (or if) people in the community think about the Poincare inequality in noncommutative geometry. In geometry, the Poincare inequality (when it exists) gives a bound on a ...

**2**

votes

**1**answer

183 views

### What is the logarithmic derivative of an (intertwining) operator?

The constant term of the Eisenstein series (for an adele group $GL_2$, say) contains an intertwining operator, often written as $M(s)$. In the form given in Gelbart-Jacquet's Corvallis paper, for ...

**1**

vote

**0**answers

150 views

### A “coarse” version of position operator and its properties

I am wondering if someone has ever studied the following operator in the context of quantum mechanics:
$$Q : L^2(\mathbb{R}) \to L^2(\mathbb{R})$$
$$(Q f)(x) := s(x) f(x),$$
where $s(x)$ is the ...

**5**

votes

**0**answers

108 views

### When is an inner derivation a Fredholm operator?

Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...

**0**

votes

**0**answers

95 views

### What are the properties of this linear operator?

Suppose $f(x)$ is a function which satisfies the following condition:
$$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$
Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...

**5**

votes

**1**answer

120 views

### A question about uniformly bounded semigroups

Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there ...

**0**

votes

**2**answers

336 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

**0**answers

71 views

### Comparison between operators

I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any ...

**0**

votes

**2**answers

91 views

### Does positivity preserve compactness? [closed]

Suppose $A$ and $B$ are operators on a (separable) Hilbert space $H$ and $A \leq B$. Is it true that if $B$ is compact then $A$ is compact too? If not, could you please show a counterexample?

**2**

votes

**2**answers

208 views

### Lower bounds for norms of commutators

For various reasons I became interested in bounds on the norm of commutators of operators. For instance, if $B(H)$ is the algebra of bounded operators on a Hilbert space, one may ask for a lower bound ...

**1**

vote

**2**answers

208 views

### Characterisation of compact operators

It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.
My question is ...

**1**

vote

**1**answer

105 views

### Characterisation of adjoint operators

Let $X$ be a Banach space and $X^*$ denote his dual space. Then, it is well-known that if $T$ is a bounded linear operator on $X$, then $T^*$ is a bounded linear operator on $X^*$. My question is the ...

**1**

vote

**0**answers

37 views

### Strong relative boundedness argument for the annihilation operator

Could someone please explain to me how to prove that the creation operator $a^*=-\frac{d}{dx} +x$ is the adjoint of the annihilation operator $a=\frac{d}{dx} + x$ in $L^2(\mathbb{R})$?
I would guess ...

**2**

votes

**1**answer

124 views

### Is this operator trace class?

Let $T:H\to H$ be a compact operator on a complex Hilbert space.
Assume that
$$
\sup_{(e_j)}\sum_j\left|\langle Te_j,e_j\rangle\right|<\infty,
$$
where the supremum extends over all orthonormal ...

**3**

votes

**1**answer

107 views

### Integral representation of the resolvent of a semigroup

Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent
$$
(\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt
$$
hold for ...

**5**

votes

**1**answer

198 views

### Self-adjoint operator

Assume that $B$ is a self-adjoint operator and $\alpha\in(0,1)$. I need a reference for the following equality $$B^{-\alpha}=\frac{\sin\alpha \pi}{\pi}\int_0^\infty ...

**6**

votes

**1**answer

202 views

### Is there a nice “minimum” of two symmetric operators?

Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt.
...

**0**

votes

**1**answer

172 views

### Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book
Singular Integrals and Differentiability Properties of Functions
that HT, when understood as a ...

**0**

votes

**1**answer

134 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 ...

**7**

votes

**1**answer

271 views

### Realisation of noncommutative torus

One of the most basic examples in noncommutative geometry is the so called noncommutative torus to be denoted by $\mathbb{T}_{\theta}$. As far as I know, there are several equivalent constructions of ...

**3**

votes

**1**answer

226 views

### When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...

**5**

votes

**1**answer

152 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.)
...

**10**

votes

**3**answers

226 views

### Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?

Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...

**0**

votes

**1**answer

105 views

### Positive kernel property

Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let
$f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that
$$\forall x\in [0,1],\quad
f(x)=\int_0^1 k(x,y) g(y) ...

**4**

votes

**1**answer

179 views

### Examples of special isometries

Are there examples of (distinct) Hilbert spaces $H_1$=$(H,\langle\cdot,\cdot\rangle_1)$, $H_2 $=$(H,\langle\cdot,\cdot\rangle_2)$ and a linear operator $V: H_1\to H_2$ such that $V^n: H_1\to H_2$ is ...

**4**

votes

**0**answers

50 views

### Strictly convex renormings making power bounded operators into contractions

Let $X$ be a Banach space and let $T$ be a power bounded linear operator on $X$ (i.e. $\sup_{n\ge0}\|T^n\|_{op}<\infty$). We can of course define an equivalent norm $\|\cdot\|'$ on $X$ so that ...

**1**

vote

**2**answers

161 views

### Self-Adjointness for Banach Spaces

Good evening. Is there a reasonable notion of being self-adjoint for the adjoint operator on Banach Spaces? Kind regards, Alex

**2**

votes

**0**answers

81 views

### Constant in Maximal sobolev regularity

We know the following evolution equation
\begin{equation}
\left\{
\begin{array}{llc}
v_t=A v+f,\\
v(0)=0.
\end{array}
\right.
\end{equation}
$A$ generates a bounded analytic semigroup on a Banach ...

**3**

votes

**2**answers

263 views

### norm of the matrix series

The goal is to obtain an upper bound for the norm of the vector
$$
\left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\|
$$
for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...

**5**

votes

**2**answers

195 views

### Is a semigroup always an exponential?

Let $H$ be a Banach space, $\mathscr{B}(H)=\{T:H\to H: \text{where $T$ is a bounded linear operator}\}$, and $S:[0,\infty)\to \mathscr{B}(H)$, a map with the following properties:
$$
S(0)=I, \quad ...

**2**

votes

**0**answers

45 views

### Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...

**5**

votes

**5**answers

405 views

### If two projections are close, then they are unitarily equivalent

Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...

**3**

votes

**2**answers

247 views

### About the trace class operators and their motivation

What is the motivation for trace class operators? Can any body suggest the most general and standard reference that includes Schatten p class operators as well.
I have following references
...

**5**

votes

**1**answer

246 views

### definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...

**5**

votes

**0**answers

198 views

### Essential unitary equivalence

Let us agree that heuristic meaning of the word "essential" is: up to compact operator. There is clear notion of unitary equivalent operators. What is the proper notion of two operators being ...

**1**

vote

**0**answers

43 views

### Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying:
\begin{align}
\Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...

**6**

votes

**2**answers

258 views

### Expression of a non-orthogonal projection in a $C^*$ algebra via an orthogonal one

A paper I'm currently reading uses the following fact. If $A$ is a unital $C^*$-algebra, $P=P^2\in A$, then there are $T, F\in A$ s.t. $F$ is an orthogonal projection ($F=F^*=F^2$) and
...

**1**

vote

**0**answers

78 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 ...

**9**

votes

**2**answers

377 views

### Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by ...