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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Prove that operator is completely continuous

Let's consider Banach space $\ell^\infty$ of bounded sequences $x = \{ \xi_n\}_{n=1}^\infty$: $$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$ Suppose matrix $||a_{i j}||_1^\infty$ specifies operator $A$ ...
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29 views

Are all cyclic representations irreducible?

I know that for a representation $\pi$ of a $^*$-algebra $\mathcal{A}$ on a Hilbert Space $\mathcal{H}$, if $\pi$ is irreducible then it is cyclic. Is the reverse implication also valid - i.e. is ...
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90 views

Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
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22 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, $T$ is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. ...
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55 views

Find the adjoint operator.

Consider the sequence space $\ell_p$ and S defined by $(1\leq p<\infty)$$$ S:\ell_p\to\ell_p:(x_1,x_2,x_3,\ldots)\mapsto(0,x_1,x_2,\ldots) $$ Find the $S^*$ operator.
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73 views

definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
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49 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
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44 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
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43 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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84 views

Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
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19 views

Polynomial generator

If we let $\alpha$ be a multiindex, can we generate any polynomial in $\eta$ with coefficients as multiples of $\kappa$ $$ D_z^{\alpha}\text{exp}(i(\kappa(z)-\kappa(x)-\kappa'(x)(z-x))\eta)|_{z=x} $$ ...
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64 views

Question about compact operators

I would like to prove the following, Let $X$,$Y$ be infinite dimensional Banach-Spaces and $T$ a compact, linear and bounded operator. Then there exists a sequence $(x_n)_{n\in\mathbb N}$ with ...
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19 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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19 views

Prove that $A\int_0^\infty S(t) u dt=\int_0^\infty S(t) A u dt$ if A is a closed operator

From Wikipedia: Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still ...
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54 views

derivation of divergence from nabla operator

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the ...
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24 views

Verification of a contraction

Let $A\colon \text{dom}(A) \to \mathcal{H}$ be a densely defined symmetric operator on a Hilbert space $\mathcal H$. The symmetry implies that $$ \|(A + i)f\|^2 = \|Af\|^2 + \|f\|^2 \quad \text{for ...
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26 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
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36 views

$K(X,Y)$ is closed in $B(X,Y)$

I tried to prove that the set of compact operators $K(X,Y)$ is a closed subset of the set of bounded operators $B(X,Y)$ where $X,Y$ are Banach spaces. Please can you tell me if my proof is correct? ...
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29 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
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30 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
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27 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
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39 views

Smoothness of solutions to Fredholm integral equation

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel ...
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Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
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32 views

Boundedness of a closed operator

Can I get any help with this problem: Let $X, Y$ be Banach spaces, let $D$ be a subspace of $X$, and let $A \colon D \to Y$ be a closed linear operator. If $D$ is a closed subspace of $X$, ...
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20 views

Unital free semigroup and operators

Let $F_n^+$ be a unital free semigroup generated by $1,...,n$. Let $\alpha= i_1...i_k$ where $i_1,...,i_k\in \{1,...,n\}$ and put $T_{\alpha}:=T_{i_1}...T_{i_k}$ where for any $ i_j\in \{1,...,n\},~ ...
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28 views

Comparison of 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 ...
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14 views

Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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44 views

Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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Examples of operator theory on Hilbert space

$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$ $(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ ...
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How to find all the eigenvalues of a positive operator whose eigenvectors are positive semi-defintie?

A linear operator $T:\mathcal{H}_n\rightarrow \mathcal{H}_n$ is said to be positive if $T(\mathcal{P}_n)\subset\mathcal{P}_n$ where $P_n$ is the set of positive semi-definite matrices. For a positive ...
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51 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
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can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
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Derivative on a function of tensor products

Assume I have defined an operator $A \otimes B$ on a $H \otimes L^2(\mathbb R^d)$ where $H$ is a Hilbert space as in Reed/Simon p. 299. $A$ is an operator on $H$ and $B$ is an operator on $L^2(\mathbb ...
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39 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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35 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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40 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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60 views

properties of integral operator $x^{-1}\int_0^xf(x,y)v(y)dy $

here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy $$ for any $x \neq 0$ otherwise $[T(v)](0) := ...
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38 views

Commutating operators modulo compact operator

Let $X$ be a Banach space. I want find an example of a closed densely defined operator $A$ and a bounded operator $B$ such that $AB - BA = K$ where $K$ is a compact operator.
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77 views

find the eigenbasis of unitary transformation

$U$ is $n\times n$ unitary matrix, with orthogonal eigenbasis $v_1, \ldots v_n$ we construct a linear transformation: $T_U(X) = XU$ with the inner product $\langle A, B \rangle = \text{tr}(A^*B)$ I ...
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Notation question in Majda and Bertozzi's “Vorticity and Incompressible Flow”

On pg 2, the fluid velocity in the Navier-Stokes system of equations is noted as: $v(x,t) \equiv (v^1, v^2, \ldots, v^N)^t$, where I am assuming that the velocity vector field is time-dependent. The ...
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When does a *-algebra have an approximate identity

I know that a *-algebra does not always have an approximate unit. When does a *-algebra have an approximate identity? Can we characterize that *-algebras which are not uniform and weak closed but ...
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186 views

Half Laplace operator

I'm curious whether a half Laplacian (or square root of Laplacian) exists. More specifically, I'm looking for an $X:C^2(\Bbb R^n)\to C^2(\Bbb R^n)$ operator such that $$\forall f:XXf=\Delta f$$ I know ...
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63 views

Three basic questions about linear operator in a Hilbert space

Just come across three questions in reading a paper. Suppose we are dealing with a Hilbert space of $L_{2}[0,1]$ and all the functions mentioned below are in $L_{2}[0,1]$. Define the operator $A$ by ...
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38 views

operator differential equation

let be the differential equation for the operator 'X' $$ \frac{dX(t)}{dt} = A(t) X(t) $$ the formal solution is the exponential operator $$ X(t)=X(0)e^{ \int_{0}^{t}A(u)} $$ of course i should ...
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39 views

Measurability of the dilatation operator

I need some help with this question: We consider the dilatation operator: $T: \mathbb{R^{+}}\to \mathcal{L}(L^p(\mathbb{R}),L^p(\mathbb{R}))$ $\;\;\;\;\;\;\delta\to ...
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Self-Adjoint Operators

If an Operator $L$ is defined as $Lu=u''$ and $a_1u(0)+b_1u'(0)+c_1u(1)+d_1u'(1)=0$ along with $a_2u(0)+b_2u'(0)+c_2u(1)+d_2u'(1)=0$, then for what values of $a_1,b_1,c_1,etc$ is the operator ...
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64 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
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64 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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144 views

Spectral theorem and projection

This should be simple, but I'm stuck. Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$. The spectral theorem says that there is a decomposition of $H$ into a direct sum for which ...
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29 views

Piecewise Continuous Fredholm Kernel

Suppose I have a ``Fredholm equation of the second kind" with kernel $$ K(x, y) = 1[x \geq u]k(x, y) $$ where k(x, y) is continuous. Let $f_n(x)$ be a smooth continuous approximation of $1[x \geq ...