# Tagged Questions

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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### A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
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### $S$ is continuous with Weak * topology from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $B(X,Y)$ [duplicate]

How to prove that prove that $S$ is weak$^*$-continuous from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $T\in B(X,Y)$ Thanks for any hints. To show that $T$ is continuous is straight forward ...
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### What is the relation between the matrix of an operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \to Y$ be a linear operator. (Then $T$ is bounded since its domain is finite-dimensional). ...
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### Prob. 5, Sec. 4.5 in Kreyszig's functional book: The adjoint of the composite of two bounded linear operators

Let $X$, $Y$, and $Z$ be normed spaces, either all real or all complex. Let $T \colon X \to Y$ and $S \colon Y \to Z$ be bounded linear operators. Let $X^\prime$, $Y^\prime$, and $Z^\prime$ denote the ...
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### Continuous inverse of an unbounded operator on a Hilbert space

Let $T:D(T)\to H$ be an unbounded densely defined operator on a Hilbert space $H$. Suppose that $T^{-1}$ is continuous, i.e. that $0$ belongs to the resolvent set $\rho(T)$ of $T$. As $T^{-1}$ exists,...
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### Norms are equivalent iff dual spaces for them are the same?

It is trivial that if we have a vector space $X$ and two equivalent norms on it than $X'_1$ -dual space (of continuous functionals) for the first norm and $X'_2$ are the same spaces. Is the converse ...
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### Proving that $-\Delta+V$ on some domain is self-adjoint

This question may look as a "proof-reading" question, but what I ask is if I correctly understand the way these concepts work, by showing how I think about them. Suppose I have the following three ...
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### operator inequality using spectral theorem

Given two densely defined unbounded self-adjoint strictly positive operator $A$ and $L$ in Hilbert space $H$ with domain $D(A) \subset D(L)$ and $\|Lx\| \leq \|Ax\|$ for all $x\in D(A)$, why do we ...
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### Boundedness of a naive integral operator

Define integral operator $J:L^2[0,1] \to L^2[0,1]$, $$Jf(x) := \int_0^x f(s) ds.$$ I am wondering if the following equivalence holds, $\|Jf\|_{L^2} \simeq \|f\|_{H^{-1}}$, where $\|\cdot\|_{H^{-1}}$ ...
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### Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
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### Characterization of orthogonal projections in terms of operator norms

I want to show the following equivalence: If $X$ is a Hilbert Space and $P\in B(X)$ (i.e. $P$ is bounded and linear) and $P^2=P$, then $$(\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1$$ I know that ...
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### Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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### $A$ is a Hermitian operator on an infinite dimensional Hilbert space and $\langle Ax|x\rangle=0$ for all $x$, prove $A=0$ without the spectral theorem

If $A$ is a Hermitian operator on an infinite dimensional Hilbert space such that $\langle Ax|x\rangle=0$ for all vectors $x$, can we prove $A=0$ without the spectral theorem? The proof seems ...
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### Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
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### If $A_j$ is an increasing family of Hermitian operators such that $A_j\nearrow A$ weakly, for $A=\mathrm{LUB}A_j$, then $A_j\rightarrow A$ strongly.

I am trying to prove the following proposition from Berberian's 'Notes on Spectral Theory': Proposition 1: If ($A_j$) is an increasingly directed family of Hermitian operators, and if the family ...
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### Inversible operator in Hilbert space

Consider $\phi\in L^{\infty}[0, 2\pi]$. Let M be operator $L_2[0, 2\pi]\rightarrow L_2[0, 2\pi]$$Mf = \phi f$$ In$L_2[0, 2\pi]$we have topological basis${e^{inx}}, n\in \mathbb Z$.$L_2[0, 2\pi] =...
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### spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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### Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
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### Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy$$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...