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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Compact operator as certain limit

Let $H$ be an infinite-dimensional Hilbert space with basis $\{e_i\}_{i=1}^\infty$. Let $P_n := \sum_{i=1}^n e_ie_i^*$, i.e. $P_n$ is the projection onto the span of the first $n$ basis vectors. Let ...
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Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
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25 views

Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes ...
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Application of the spectral mapping theorem

Let $T:L^2((0,2)\rightarrow L^2((0,2))$, $(Tx)(t):=\begin{cases} x(t+1), & 0<t<1\\ 0,& \text{elsewhere} \end{cases} $ Show that $T$ is well defined and $\sigma(T)=\sigma_p(T)=\{0\}$ ...
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Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f''$ ...
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Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
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An invertible hermitian element of a C*-algebra has a logarithm

Suppose $ A$ is a C*-algebra. Show that an invertible hermitian element of $A$ has a logarithm. ($a$ has a logarithm if there is an element $b\in A$ such that $e^b=a$) If $a\in A_+$ then it's easy ...
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is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
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Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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Is everything an operator?

For example, I have some number $\alpha$ and a function $f$. Now I multiple this constant $\alpha$ with $f$ and get $\alpha * f$. Now I claim that $\alpha$ is an operator, $f$ my eigenvector, with ...
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If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$. But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/
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Sums of two closed and closed / continuous operators

Let $X$ be a normed space and $A_j:D(A_j)\rightarrow X$ (j=1,2) linear. (i) If $D(A_1)=X$, $A_1$ continuous and A_2 closed. Do we have $A_1+A_2:D(A_1)\cap D(A_2) \rightarrow X$, $x\mapsto A_1x+A_2x$ ...
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Unitary Equivalent of Derivative in Fourier Space

It is known that for $L^2(\mathbb R)$ the operator $Tf(x) = if'(x)$ is unitary equivalent to $\hat T \hat f(\xi )= \xi \hat f(\xi) $. Where domain of T is $H^1(\mathbb R)$. Hence the Spectrum of T in ...
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Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
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For positive operators $A$ and $B$ with $A^6=B^6$ show that $A=B$

Since $A$ and $B$ are positive, I managed to show that $A^6$ and $B^6$ are positive. Now, I can use the fact that there exists a unique square root of both of those and since they're equal, their ...
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Show that the given family of bounded operators on a hilbert space form a semi group.

Suppose $A:D(A)\subset H\rightarrow H$ is a self adjoint, densly defined closed operator and it is also positive operator i.e $<Au,u>\geq0 $, for all $u\in H$ ,where $H$ is a hilbert space . ...
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Linear compact operators

Let $X$ be an infinite-dimensional Banach space, $Y$ be a Banach space, $A: X \to Y$ be a linear compact operator. Is it true that there is always a sequence $\{x_n\}\subset X$ such that $\|x_n\| \to ...
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Maximal ideal space

Let $X$ be a compact space, $x_0\in X$, and define $$A=\{\{f_n\} ; f_n\in C(X), \sup_n\|f_n\|<\infty, and \{f_n(x_0)\} \text{ is a convergent sequence} \} $$ If $\|\{f_n\}\|$ is defined as ...
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Existence fixed point

Let $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and compact valued. Consider the function $F: \mathbb{R}^n \times ...
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Invertible operators on a separable Hilbert space

Using polar decomposition or Kuiper's theorem one can show that the set of invertible operators on a separable Hilbert space $H$ is a connected subset of ${\mathcal B}(H)$. But does anyone know an ...
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A question about adjoint matrices

Let $T:V \to V $ be a linear map on complex vector space $V$ which is equipped with complex inner product $ <. , .> $ we know there exists a unique linear operator $T^* : V \to V $ such that ...
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How to understand all types of transforms as linear operators on function spaces?

I would really appreciate if someone can point to a simple general framework that can help me to understand integral transform from a generalized point of view so that there is no longer a fourier ...
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3answers
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Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
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Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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Essential self-adjointness of the Laplace operator via the Fourier transform

I'm working through some notes on showing the essential self-adjointness of the Laplace operator on $\Bbb R$ via the Fourier transform (see here) but there seems to be a little bit of liberty taken at ...
2
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Riemann manifold with unbounded Laplacian

How can one characterize a Riemann manifold the Laplacian of which is unbounded? (Equivalently, what are those manifolds on which the Laplacian is bounded? I am interested in working with its ...
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Is the injectivity of the operator equivalent to the surjectivity of its adjoint

Let $X$ and $Y$ be two normed linear spaces. Let $T:X \to Y^*$ be a linear operator (not necessarily continuous) and let $T^*$ be its adjoint, i.e. $T^*:Y \to X^*$ is defined by $ \langle T^*y,x ...
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Existence of a linear operator

Does there exist a linear operator $A:\ell^2 \to \ell^2$, $$ A(x_1, x_2, \ldots, x_n, \ldots) = (y_1, y_2, \ldots, y_n, \ldots) $$ such that $\exists x = (x_1, x_2, \ldots, x_n, \ldots)$: ...
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Compact space X is totally disconnected if and only if C(X) is generated by its projections

If $X$ is compact, show that $X$ is totally disconnected if and only if $C(X)$ as a C*-algebra is generated by its projections. My attempt: Suppose $X$ is totally disconnected, then $X=\{x_i\}_{i\in ...
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Linear operator norm

I am trying to show that these two definitions for a bounded linear operator norm on the normed linear space $X$ are equivalent: $$ \sup\{T(x)\,:\, \|x\|\le 1\}=\|T\|_*=\inf\{M>0\,:\, T(x)\le ...
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A non-continuous idempotent linear operator in a Banach space

Does there exist a non-continuous idempotent linear operator $P: X \to X$ where $X$ is an infinite-dimensional Banach space? That is, $P^2 = P$, and there is a sequence $\{x_n\}$ of elements of X such ...
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Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
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References on projectors

What are good books or articles about linear projectors in Hilbert spaces? I am mostly interested in the finite dimensional case (but anything is welcome). All about idempotents, orthogonal and ...
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Perturbation of Laplacian

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
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Proving equivalence of two definitions

Hi all I am intersted in proving the equivalence of the following two definitions of pseudomontoncity: Let $V$ be a reflexive Banach space and $K \subset V$ closed and convex. Definition 1: $A: V ...
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the sum of two unbounded normal operators

why A and B are normal?and why "0" is not closed on H1(R)?
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Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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Question about linear operator continuity

Let $A:X\rightarrow Y$ be a linear operator, $X,Y$ normed spaces. Show that a linear operator is continuous (bonded) if for every sequence $x_n\rightarrow 0$ in $X$ has a bounded image $Ax_n$ in $Y$. ...
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Does this operator equation have solutions?

Hi Math StackExchange community, I have a question that originates from a Physics problem; the question itself however is about solving an operator equation. In a particular quantum mechanical ...
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How do I prove this statement about the operator norm?

I stumbled across this equation in a paper, which may seem obvious, but I'm wondering if someone can explain why this is true? By definition of an operator norm, $$\left[(D^*D)^{-1} - ...
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Approximation Property: Characterization

As reference the german wiki: Approximationseigenschaft Problem Given a Banach space. Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$ ...
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Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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1answer
25 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a bounded operator: $$W:\mathcal{H}\to\mathcal{K}:\quad\|W\|<\infty$$ Then a partial ...
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Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
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Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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Separating and cyclic vector

Let $\{\Gamma_i , \mu_i\}_{i\in I}$ be a family of probability measure spaces and suppose $I$ is uncountable. Let $\{\Gamma , \mu\} = \prod_{i\in I} \{\Gamma_i,\mu_i\}$ be the product measure space. ...
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Irreducible representation and rank one projetion

Let $A$ be a C*-algebra with a nonzero minimal projection $e$. a - Show that if $\{\pi, H\}$ is an irreducible representation of $A$ such that $\pi(e) \neq 0$, then $\pi(e)$ is a projection of rank ...