# 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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### $Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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### Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
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### Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R},$$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
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### Different ways of decomposing an exponential map

There are many decompositions of an exponential map which has two (or more) operators in the exponent (i.e. $e^{A+B}$, where $A$ and $B$ are operators). For example, the Baker-Campbell-Hausdorff (and ...
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### Why is an operator composed with its adjoint positive and stricly positive when it's invertible?

Let $V$ be a (complex) finite vector space equiped with an inner product and $T$ an operator on V. We say $T$ is positive if: $$\langle T(v), v \rangle \geq 0$$ for all $v$ in $V$. We say $T$ is ...
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### Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of ...
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### Characterization of compact operators by their spectra

In any functional analysis book there is usually a section devoted to the study of the properties of the spectrum of compact operators. Is there any spectral characterization of compact (self-...
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### Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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### $\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
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### When is orthogonal projection compact? [duplicate]

Let $M$ be a closed subspace of a Hilbert space $H$. Let $P$ be the orthogonal projection on $M$. I was told to find the eigenvalues and eigenvectors of $P$ and moreover say when it is compact. Since ...
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### Simple norm inequality

Trying to follow the comments to this question I am struggling very much to understand how to simplify $\|Ax\|_2=\sup_{\|x\|_2=1}\sqrt{\sum_i(\sum_ja_{ij}x_j)^2}$ to arrive at an $x$-free bound. Can ...
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### Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
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### $(k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $(k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
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### Proving an operator is compact exercise

Suppose $(a_{ij})_{i,j\in \mathbb N}$ satisfy $\sum_{i,j}|a_{ij}|^2<\infty$ and define $A:\ell ^2 \rightarrow \ell ^2$ by $(Ax)_i)=\sum _j a_{ij}x_j$. I need to prove $A$ is compact. Unfortunately,...
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### Trace norm of a triangular matrix with only ones above the diagonal

For $n\in\mathbb N^*$, we consider the triangular matrix $$T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,.$$ ...
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### On linear homotopy of operators

Let $F$ be an isomorphism of euclidian space $E$, with orthonormal basis $\{e_1,\ldots e_n\}$. Let $F'$ be orthogonalised $F$. Is any operator $F_t$ from linear homotopy of $F$ and $F'$ an ...
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### Determining whether equality $\|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
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### Integral operator convergence study in $L^2(\mathbb{R})$

Exercise I want to study in $\mathcal{H}=L^2(\mathbb{R})$ the convergence of $$A_nf(x)=\log n \int_\mathbb{R} \frac{1}{1+n(x-y)^2}f(y)dy.$$ Solving a) Pointwise convergence : For $n \to \infty$ ...