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 $(U,〈\;⋅\;,\;⋅\;〉),H$ 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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39 views

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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24 views

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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40 views

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 $(H,\langle\;\...
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13 views

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 $p=\...
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16 views

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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22 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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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 $P[A]=\bigcup_{n=1}^\infty\...
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37 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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24 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 $\mathcal{K}\left(\mathcal{H}\right)...
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1answer
30 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 ...
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29 views

Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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23 views

Given a normed space $X$ and $A:X\to\mathbb R$, how can I compute the second Fréchet derivative of $f(t):=A(x_0+th)$ for some $x_0,h\in X$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a Banach space and $A:X\to\mathbb R$ be Fréchet differentiable, i.e. $\exists{\rm D}A:X\to\mathfrak L(X,\mathbb R)$$^1$ with $$\lim_{\left\|h\right\|\to 0}\frac{\...
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22 views

semifinite von Neumann algebra, spectral projection, trace

Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, ...
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1answer
19 views

Does a polynomial function on spectrum uniquely define polynomial on operator?

Let $X\subset\mathbb C$ be a compact set, let $T$ be a bounded operator with its spectrum contained in $X$, let $P$ be a polynomial. Is it true that whenever $P=0$ on $X$ then $P(T)=0$?
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Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
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1answer
20 views

spectral projection

Let $T$ be a self-adjoint operator on a Hilbert space $H$. $P$ is a projection on $H$. Let $E^{|PTP|}(1,\infty)$ be a spectral projection of $|PTP|$. My question is: whether $E^{|PTP|}(1,\infty) \le P$...
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44 views

The density of $C^1[0,2\pi]$

I am not sure if the inclusion $\{f \in AC[0,2\pi]: f(0)=f(2\pi)=0\}\subseteq \overline{\{f \in C^1[0,2\pi]: f(0)=f(2\pi)=0\}}.$ Here $C^1[0,2\pi]$ is the set of continuously differentiable functions ...
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30 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
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2answers
35 views

positive operator, projection, Hilbert space

Let $T$ be a positive operator on a Hilbert space $H$. Let $P$ be a projection on $H$. Then, it is well-known that $PTP$ is also positive. My question is: whether $T\ge PTP$?
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Self adjoint and symmetric operator

I am wondering whether for an operator defined on a real Hilbert space to be positive we need to show that it is self-adjoint at first. It seems to me that they are two different property and can be ...
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25 views

If $U,H$ are Hilbert spaces, $Q$ is an operator on $U$ and $U_0:=Q^{\frac 12}U$, find an expression for the norm of Hilbert-Schmidt operators $U_0→H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for ...
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If $Q$ is an operator on a Hilbert space $U$ and $(e_n)_{n\in\mathbb N}$ is an ONB of $U$ with $Qe_n=λ_ne_n$, then $Q^{-1}e_n=\frac 1{λ_n}e_n$

Let $U$ be a Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\...
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Is there a way to compare the following two operator norms?

Suppose that $K:C([0,1])\to C([0,1])$ is a continuous operator both with respect to $L^2$ and $L^\infty$ norms. Consider the following operator norm $$\sup_{\|f\|_2\leq 1}\|Kf\|_\infty$$ where $\|.\|...
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Show $(Au)(x)=v_1(x) \langle u , v_1 \rangle+ v_2 (x) \langle u, v_2 \rangle$ where $(Au)(x)=\int^\pi _0 2u(y) cos \bigg (\frac{x-y}{2} \bigg) dy$

Let $A$ be a linear operator defined on $L^2([0, \pi])$ by $$(Au)(x)=\int^\pi _0 2u(y) cos \bigg (\frac{x-y}{2} \bigg) dy$$ Where $0 \leq x \leq \pi$ I am trying to show that $(Au)(x)=v_1(x) \...
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50 views

When is this matrix positive semidefinite?

Let us fix dimension $n$. Consider the $n \times n$ matrix \begin{equation} S_n=\begin{bmatrix} 1 & z & z & \cdots & z \\ \bar{z} & 1 & z & \cdots & z\\ \bar{z} & \...
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Domain of the closed extension operator is not the entire space?

Given a Banach space $X$, and a densely define linear operator $A:D(A)\subset X \rightarrow X$, we define the graph $$G_A= \{ (x,Ax) | x\in D(A)\}$$ which is a linear subspace of Banach space $X\...
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Self adjoint operators on Hilbert spaces are bounded

I think I have a proof that if $A: H\rightarrow H$ is a self adjoint operator on a Hilbert space $H$, then $A$ is bounded: We can use the closed graph theorem. Let $x_n \rightarrow x$ and $Ax_n \...
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Eigenfunction of a selft-adjoint operator?

Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by $...
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Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$ \langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H; $$ strongly monotone if there exists ...
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When can I Taylor expand a function of an operator?

1-) Is the expression $f(A) = \sum_n \frac{f'(0)}{n!}(A)^n$ always meaningful for any diagonalizable linear operator $A$ and for any analytic function $f$? This seems strange to me because then I ...
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1answer
43 views

A property of the Volterra operator

I was reading a paper and I came across the Volterra operator $$(Vf)(x)=\int ^x _0 f(t) dt$$ And its adjoint $$(V^*f)(x)=\int^1 _x f(t) dt$$ It also says that a simple and useful identity is $$V+V^*=...
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Strong Convergence of Fredholm Operators, as used in Callias' proof of his index theorem

In his paper Axial Anomalies and Index Theorems on Open Spaces, Callias provides a wonderful index theorem $$\mathrm{index}(L)=\lim_{z\to0} \mathrm{Tr}B_z\quad\text{where} \quad B_z=\frac{z}{L^\dagger ...
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Understanding Operator Norm of Matrices

Let $X$ denote the vector space of $n\times n$ complex matrices. To every matrix $A\in X$ one can associate two operator norms: Thinking of $A$ as a map $A\colon \mathbb{C}^n\to \mathbb{C}^n$ or $A\...
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Why is it true that the multiplication operator in a reproducing kernel Hilbert space is always continuous?

In my functional analysis I was met with this seemingly trivial theorem on RKHS If $ \mathbb{H} $ is a reproducing Kernel Hilbert Space and we have a multiplier $ \phi $ meaning it satisfies $ \...
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Is the set of adjoint operators weak* closed?

Suppose we have a Banach space $X$ and a net of bounded operators $(T_\gamma)$ on $X$ such that $T_\gamma^*\to S$, for some bounded operator $S$ on $X^*$, where the convergence is with respect to the ...
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Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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1answer
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Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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50 views

Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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Spectrum of an Operator on a Banachspace

Claim: Let $A$ be a bounded linear operator on a Banachspace $\mathfrak{X}$. Denote $\sigma(A)$ as the spectrum of A. Let $\lambda$ be a point in the boundary of the $\sigma(A)$. Then there exist a ...
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Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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Proof that a linear operator is continuous.

Could somebody please verify the following proof I have attempted? It seems too simple so I am worried I have done something wrong.. Many thanks Let $T:(X,\|.\|_X)\to (Y,\|.\|_Y)$ be a linear map ...
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Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius $...
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1answer
22 views

If a contraction and its adjoint converge to zero both does that mean the contraction satisfies $ ||Th|| < h $

I just met this in my functional analysis on contractions which got me stumped: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \leq ...
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33 views

Does a contraction converging in power series necessarily lead to the operator being a proper contraction?

I was recently met with this in my functional analysis class on which I am stuck: Let $ \mathbb{H} $ be a Hilbert space and let T be a contraction operator on $ \mathbb{H} $ (meaning $ ||T|| \...
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1answer
33 views

Index of a derivative operator on a circle

Let $D: C^{1}(S^{1}) \rightarrow C(S^{1})$ be an operator defined as $D(f)=f'$. I would like to find its index (on the road proving that it's a Fredholm operator). First, if $f \in ker(D)$, then $f$...
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43 views

Two sequences of operators on Hilbert Space

Let $H$ is some Hilbert Space, and $a_n,b_n \in B(H)$ is sequences of some operators on it. We know, that $a_n b_n$ converges to $v$ by norm. We also know, that all $b_n a_n$ are strictly positive. ...
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35 views

For $\{T_n\}$ and $T$ positive and self-adjoint, show $T_n \stackrel{SR}{\to} T$ iff $(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$

For $\{T_n\}$ and $T$ positive and self-adjoint, show $T_n \stackrel{SR}{\to} T$ (i.e. $T_n \to T$ in the strong resolvent sense) iff $(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$ (i.e. $(T_n + I)^{...
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1answer
17 views

Index of a differential operator

Let's consider an operator $D: C^{m+n}[a, b] \rightarrow C^{m}[a, b]$, defined as $D(y(t)) = y^{(n)}+a_{n-1}y^{(n-1)}+\ldots+a_{1}y'+a_{0}$, $a_{k} \in C^{m}[a, b]$. I would like to prove that it is ...