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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Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} ...
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Compactness of a bounded operator $A: l_1 \to l_1$

Let $l_1$ denote the space of absolutely summable sequences and $B(l_1,l_1)$ denote the space of all bounded linear operators from $l_1$ to $l_1$. I am trying to solve the following question Let $A ...
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Functions over a $C$ vector space with geometric importance. (How to find the basis?)

Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex ...
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Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...
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Minimality of the Spatial C*-Norm

Given C*-Algebras $A,B$ and $x \in A \otimes_* B$ show that: (a) If $(\tau \otimes_* \rho)(x)=0$ for all $\tau \in A_+^*$ and $\rho \in B_+^*$, then $x=0$. PS: $\tau \in A_+^*$ means that $\tau$ is ...
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Equivalent ways to study perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates a $C_0$ ...
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Spectrum of a polynomial operator?

Let us have $A: l^2 \to l^2, A \in B(l^2)$. $$A(\delta_n)=3 \delta_{n}+i \delta_{n+1}$$ What is the spectrum of $A$? My approach: We can write down $A$ in a better form: $$A=3I - iR$$, where $I$ ...
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48 views

Aproximating positive elements in inductive limit of C* algebras

Let $\{A_i,\Phi_{ij} \}_{i\in \mathcal{I}}$ a directed system of C* algebras and $A:=\varinjlim A_i$ its limit. I know that if $x\in A$ is self-adjoint, it can be approximated with another ...
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Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
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Operator groups

In $H := L^2(\mathbb{R}, \lambda)$ Hilbert-space, the following two, one-variable operator groups are given: $$(U_s f)(x):=f(x-s)$$ $$(V_s f)(x):=e^{is x} f(x)$$ $f \in H, s \in \mathbb{R}$. a, ...
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A question on Bounded Approximation property

Let $V$ be a Banach space and we say that $V$ has the $C$-BAP if there exists a net of bounded finite rank operators $T_\alpha$ in $B(V,V)$ and a constant $C$ such that $\|T_\alpha\| \leq C$ for each ...
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Operator continuity on Hilbert space

Let $A: H \to H$ be a linear operator on Hilbert space $H$, and let $\{\alpha_n\}_{n = 1}^{\infty} \subset \mathbb{R}$ converges to nonzero number. Prove that if the series $\sum_{n = 1}^{\infty} ...
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If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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Example for injective and surjective bounded and unbounded operator

I am looking for some bounded and unbounded densely defined operators on a real Hilbert space $H$, let say $A:D(A)(\subset H)\to H$, that are one-to-one but they are not onto. I am wondering whether ...
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Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
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Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset ...
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Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
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inverse of operator

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial ...
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15 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
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Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for ...
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Operator Sum: Selfadjoint

Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}A\to\mathcal{H}:\quad A=A^{**}$$ Does it follow that: $$S:=\overline{A+A^*}:\quad S=S^*$$ (Rigorous proof?) Densely ...
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Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle ...
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Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_{n\in\mathbb N}$, what's the derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle ...
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51 views

Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all ...
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Neumann Poincare operator maps $L^2$ in itself

How can I show that the Neumann-Poincare operator $$ K_{\partial \Omega}[\phi](x) = \int_{\partial \Omega} \dfrac{(x-y) \cdot \nu(y)}{|x-y|^d} \phi(y) \ dy $$ maps $L^2(\partial \Omega)$ in itself (if ...
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Hilbert space …

i got the following operator. Let H be a Hilbert space, $(\lambda_n)_{n \in \mathbb{N}}$ a bounded sequence in $\mathbb{K}$ and $T^: H \to H, x \to \sum_{n\in\mathbb{N}} \lambda_n \left<x, ...
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Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
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Form of pure states on $M_n(\mathbb{C})$

Related question: The form of the states on an algebra of $n\times n$ matrices with complex entries I have tried to show that pure states on $M_n(\Bbb{C})$ are of the form $\phi(A)=Tr(\rho A)$, just ...
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Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ ...
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Tridiagonal matrix inner product inequality

I want to show that there is a $c>0$ such that $$ \left<Lx,x\right>\ge c\|x\|^2, $$ for alle $x\in \ell(\mathbb{Z})$ where $$ L= \begin{pmatrix} \ddots & \ddots & & & \\ ...
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1answer
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non direct sum decomposition using Fredholm operators

I am reading a paper on PDEs and at some point the author uses an argument that I cannot understand very clear, it seems to be elementary but I do not get it. Let $P$ be a Fredholm operator (i.e. ...
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Continuity condition in locally convex F - spaces

It is well known that if $X$ and $Y$ are Banach spaces, then if the linear map $T:X→Y$ satisfies the condition $f∘T ∈ X^{*}$ for every $f∈Y^{*}$, then T is bounded. Is the conclusion still true under ...
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Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces ...
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Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$

Suppose that $F(s)=\mathcal{L}\{f(t)\}$ (Defined as the usual Laplacian operator). Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$. ...
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If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$ $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$ How ...
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Sequence of bounded Operators (Is this a counterexample?)

I've to proof the following statement Let $X,Y$ be to banach spaces and $(T_k)_{k \in \mathbb{N}} \subseteq L(X,Y)$ a bounded sequence of bounded linear operators. Further it exists a dense subset ...
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Adjoint Operator to the Derivative

Let $V \subset \Bbb R[X]$ be the Vectorspace of all Polynomials of degree $\le 3$. The inner product on $V$ is defined as follows: $$\langle f,g \rangle:=\int^1_{-1}f(t)g(t)dt$$Let $L:V \to V$ be the ...
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Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
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If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let ...
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1answer
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Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
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18 views

Following integral is in the domai of the operator or not.

Let $\{T(t):t\geq 0\}$ be a $C_0$ semi group on a Banach space $X$ and let $A:D(A)\to X$ be its infinitesimal generator. We know that for $x\in X$, $\int_{0}^{t}T(s)xds\in D(A).$ Can we conclude ...
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Need hint to show that an operator is compact [closed]

Let H be a separable Hilbert space with the basis $\{e_n\}$. If A is an operator defined by $$Ae=\frac{1}{n} e_n$$ then show that A is compact. I just need a hint how to solve such type of problems.
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The necessity of defining the stable equivalence in the construction of the Grothendieck group $K_0$

I am confused about the process of the construction of the Grothendieck group $K_0$ in Murphy's $C^*$-algebras and operator theory section 7.1. Let $A$ be a $*$-algebra and ...
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30 views

Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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19 views

Fredholm index in Calkin Algebra

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space, let $\mathcal{B}\left(\mathcal{H}\right)$ be the Banach algebra of bounded linear operators and ...
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1answer
22 views

Compact Operator with Infinite rank Doesn't have a Closed Image

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. Claim: A compact operator $T$ which has infinite-rank has an image that isn't closed. I'm trying to prove this claim but I'm ...