Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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If A unitary matrix and orthogonally diagonalizable why there is a basis in whichthe linear trans. matrix is diagonal?

If $A$ is a $n\times n$ unitary matrix (above the complex field) and is orthogonally diagonalizable, why does it mean that the is an orthonormal basis $\mathbb C$ in which the matrix that represent ...
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40 views

Operator Theory Textbook Question

I read the following excerpt in my course textbook: Now, I'm led to believe that $P:X\rightarrow X$ as above is bounded iff $M$ and $N$ are both closed. I understand the if direction, but I can't ...
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21 views

Question about operator algebra

I'm not certain about the rules of operator algebra, and I am wondering if these statements are equivalent $$\left(z^2\frac{d}{dz}-2z\right)\cdot\left(z^2\frac{d}{dz}-2z\right)=$$ ...
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22 views

Positive elements below projections

Let $a$ be a positive element in $A$, where $A$ is a $C^*$-algebra. Let $p\in A$ a projection and suppose $a\leq p$. Is it true that $ap=pa$? If yes, shouldn't we have $ap=pa=a$, since ...
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45 views

Adjoint of unbounded Operators: Product and Sum

When precisely does equality hold for sum and product: $$S^*+T^*\subseteq (S+T)^*$$ $$S^*T^*\subseteq (TS)^*$$ So far I checked that for the sum the seemingly weaker condition implies the stronger ...
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105 views

Exercise about compact operator.

In $X=\ell^p$, $p\in[1,\infty]$ we consider: $$ T(x_1,x_2,x_3,\ldots)=(0,x_1,0,x_3,\ldots) $$ Prove that $T$ isn't a compact operator and that $T^2$ is a compact operator. I think I solved the second ...
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33 views

orbits of a bounded linear operator

What interesting or strong results are there concerning orbits of an operator and invariant susbpaces (either in banach or hilbert space)? Obviously, I know that an operator T has an invariant ...
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86 views

Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
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16 views

partial differential operators

As we know, there exists a semigroup for partial differential operators $A = \sum_{i,j=1}^N D_i(a_{ij}(\cdot)D_j)$, see (Klaus-Jochen Engel, Rainer Nagel, one-parameter semigroups). My question is ...
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1answer
88 views

Bounded Inverse Theorem

$A$ is a bounded linear operator from $X$ to $Y$ (both Banach spaces). Show that if there exists $k > 0$ such that $\|Ax\| \geq k\|x\|$, for all $x$ then $\operatorname{range}(A)\,$ is closed. My ...
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168 views

Gateaux and Frechet derivatives and related notions

Let $X$ and $Y$ be normed real vector spaces, and $f : X \to Y$ a map. Let's say that: G) $f$ is Gateaux differentiable at $x_0 \in X$ if for all directions $v \in X$ the limit $f'(x_0)(v) := ...
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1answer
22 views

unbounded operator with no invariant subspace

Is there an example of an unbounded operator on a hilbert space with no invariant subspace? Or is there some other reason why we are only interested in bounded case.
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49 views

Condition for vector to be in the domain of unbounded operator.

Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit $$ \lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\| $$ exists and is finite. Does ...
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84 views

Gateaux Derivative.

Let $X$ be a Banach algebra. For $f\in X$, an operator $F_t$ is defined as \begin{equation} F_t(f)=\begin{cases} \Big\{\frac{f^t}{t(t-1)}, & t\neq 0,1; \\ \\ ...
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84 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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1answer
46 views

Commutating operators and tensor products

I have this lecture slides about commutators and tensor products, but there is one part that I don't understand: The operators and are commuting operators on the tensor product and their sum has ...
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36 views

Exercise about linear operator

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ ...
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47 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
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94 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
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38 views

Linear Operator: Boundedness

I'm stuck at: $\sup_{\overline{B_1}}\lVert T x\rVert\leq\sup_{B_1}\lVert T x\rVert$? For sure it holds: $\sup_{B_1}\lVert T x\rVert=\sup_{B_1\setminus\{0\}}\lVert x\rVert\lVert T \frac{x}{\lVert ...
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1answer
30 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
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0answers
191 views

Eigenvalues of self-adjoint eigenvalue problem

I am stack with the following problem: Consider the following eigenvalue problem $$ u \in H_B(0,1), \; \langle Lu, Lv\rangle = \lambda (\alpha \langle u, v\rangle + \langle u', v'\rangle) \; \forall ...
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$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

Denote by $H^s(\mathbb{R}^n)$ the Sobolev space on $\mathbb{R}^n$ of order $s \in \mathbb{R}$ and recall that we have $H^s(\mathbb{R}^n)^\ast \cong H^{-s}(\mathbb{R}^n)$ for the dual space of ...
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83 views

How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
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157 views

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
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1answer
34 views

Closed range assumption in definition of Fredholm operators

There are two definitions of Fredholm operators (on a Hilbert space) that are commonly used. The first is that $\dim\ker T<\infty$ and $\dim\,\mathrm{coker} T<\infty$. An argument using the open ...
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51 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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81 views

Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2} $ measurable sets, $\displaystyle 1 < p,q < \infty $ and consider the measurable function ...
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55 views

The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...
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Is that operator positive-definite?

Let's consider the integral operator $\phi(x) = \int\limits^1_0\psi(y)\ln\Bigl(\Bigl|\frac{\sqrt{1-x^2}+\sqrt{1-y^2}}{\sqrt{1-x^2}-\sqrt{1-y^2}}\Bigr|\Bigr)\,dy$. How to check is this operator ...
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32 views

Decomposition of Partial Isometry

I'm reading a paper and I don't understand how the operator is being decomposed. I've tried reading about different types of decomposition but nothing I read seems relevant: (Let $\mathscr{H}$ be a ...
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30 views

The product of two projections is 0

I'm reading a paper and the paper seems to think the following is obvious: Let $S$ be a semigroup of partial isometries. Let $R = \{ E \in P(S) \cup Q(S) : E$ is minimal in $P(S) \cup Q(S)$ and for ...
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54 views

Properties of an additive mappings which preserves projections

Let $A$ and $B$ be two $C^{*}$-algebras and $\Phi:A\longrightarrow B$ be an additive map which satisfies $\Phi(0)=0$, $\Phi(I)=I$ and $\Phi$ preserves projections, (i.e, $\Phi(P)=Q$ where $Q$ is also ...
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204 views

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators ...
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112 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
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74 views

Proving that two operators are equal

So I'm trying to prove that there is an equivalence between $\langle \psi\mid T\varphi\rangle=\langle\psi \mid S\varphi\rangle$ and $\langle\varphi \mid T\varphi\rangle=\langle\varphi \mid ...
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Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
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2answers
54 views

Distance between unilateral shift and compact operator

We have $S\in\mathbb{B}(\mathcal{H})$ (where $\mathbb{B}(\mathcal{H})$ is algebra of bounded linear operators in Hilbert space) and $S$ is unilateral shift. Compute ...
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1answer
97 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
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norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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1answer
57 views

A question on operator theory

Let $T$ be a quasinilpotent operator acting on a separable Hilbert space $H$. Fix a vector $x$ in $H$ such that $[T^n x]=H$ (the closed span of the orbit is $H$), and a hyperplane $Z\subset H$. Can we ...
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20 views

An operator with infinite deficiency index

I'm looking for a simple example of an operator with infinite deficiency index .
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51 views

The anti-symmetrization and simetrization operators are mutually orthogonal

For each vector $x=(x_1,\dots,x_n)$ of an $n$-dimensional vector space $V$, and for each permutation $s$ of the symmetric group on the $n$-element set $S_n$, put $s(x)=(x_{s(1)},\dots,x_{s(n)})$. Then ...
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28 views

Conditions for an Operator to Map Onto

Let operator $A[f(x)]=g(x)f(x)$ such that $A:C[a,b] \rightarrow C[a,b]$. I'm trying to think of the necessary and sufficient conditions needed on $g(x)$ such that the map is onto. Obviously it needs ...
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1answer
42 views

Multiplicative operator from L1 to L1 is given by an L_inf function

Problem: Let $\phi :X\rightarrow \mathbb{C}$ be a measurable function with respect to a measure space $(X,\mu)$. Suppose that $\phi f\in L^1(X,\mu)$ whenever $f\in L^1(X,\mu)$ and define $M_\phi ...
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113 views

Positive unbounded operators

Let $T$ be an operator in $H$. We say self adjoint $T$ is positive iff $(\forall x\in H)\langle Tx,x\rangle \geq 0 $. As in the case of bounded operators, it is true that a self-adjoint operator $T$ ...
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1answer
58 views

How to prove that the operator $(\lambda I-A)^{-1}$ exists?

Let $A:H^1(\mathbb{R})\to L^2(\mathbb{R})$ be the operator given by $Aw=w_x$, where $w_x$ denotes the weak derivative of $w$. I need help to prove that $(\lambda I-A)^{-1}$ exists and is bounded for ...
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75 views

Integral Operator Theory on $L^2[0,1]$

Let K be the integral operator on l^2[0,1] defined by itex(t) = \int_0^t (t-s)f(s)\,ds[/itex] where 0\leq t\leq 1 Show that ||K|| <1 and that tex(t)= \int_0^t ...
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2answers
57 views

Self-adjoint Hilbert Space operators

Let $H$ be a Hilbert space and $T$ is a self adjoint continuous operator in $\mathcal B(H)$. Show that $\|T^{2^{k}}\| = \|T\|^{2^{k}}$. Does this equality hold for all operators? Now it is clear that ...
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56 views

$U$ linear and bounded, is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$

"Let $H$ and $G$ be Hilbert spaces and let $U:H \rightarrow G$ be a bounded operator. Prove that $U$ is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$." I have denoted $U^*$ to be the ...