**2**

votes

**1**answer

175 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

146 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

106 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

87 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

118 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

292 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

90 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

198 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

205 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

104 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

98 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

197 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

199 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

160 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

132 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

255 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

202 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

145 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.)
...

**9**

votes

**3**answers

207 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

103 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

173 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

159 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

261 views

### norm of the matrix series

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

**5**

votes

**2**answers

193 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

376 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

234 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
...

**0**

votes

**0**answers

112 views

### Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21, in a note by professor Terrence Tao on his own blog
http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/
Exercise 21 Suppose we are in the situation ...

**5**

votes

**1**answer

224 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 âˆ’ \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...

**5**

votes

**0**answers

197 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

247 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

374 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 ...

**1**

vote

**0**answers

188 views

### Compact integral operator

I have a question regarding compact integral operators on $L^{2}({\Omega})$ with $\Omega$ a bounded domain in $\mathbb{R^{n}}$ Suppose we are given $T$ from $L^{2}(\Omega)$ to $L^{2}(\Omega)$ as ...

**0**

votes

**0**answers

70 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?

**2**

votes

**2**answers

114 views

### Given a subdomain of GL(n), when is the map from matrices to their matrices of eigenvectors a diffeomorphism?

I'm wondering if there are any general conditions on a subdomain of $GL(n)$, which would guarantee that the map from a matrix to its matrix of eigenvectors is a diffeomorphism.
For example, given a ...

**2**

votes

**0**answers

59 views

### series representation for *un*bounded perturbations of semigroup generators

Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then ...

**2**

votes

**1**answer

254 views

### Question on structure of von Neumann algebras, clarification in Conway's “A course in operator theory”

I was reading the section on the structure of type I von Neumann algebras in John B. Conway's "A course in operator theory" and a few questions about certain definitions and references arose, I was ...

**3**

votes

**2**answers

193 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

199 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:
($*$) ...

**9**

votes

**2**answers

794 views

### How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e.
$$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$
Then is ...

**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, ...

**4**

votes

**1**answer

265 views

### extending compact operators to c0

Lindenstrauss has the following paper: http://www.ams.org/journals/bull/1962-68-05/S0002-9904-1962-10787-3/S0002-9904-1962-10787-3.pdf
I would like to see the proof for the following theorem (from ...

**1**

vote

**0**answers

85 views

### A generalization of strictly upper triangular matrces

Let $A = [a_{ij}]_{n\times n}$ be a real matrix with the property $a_{ij}a_{ji} = 0$. What can be said about the eigenvalues of$A$ ? I want to know when $A$ is non-singular and when $A$ is nilpotent. ...