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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Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
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Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $\int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
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Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
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Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by \...
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continuity of a function and net convergence

The following is a statement and its proof in the Banach Algebra Techniques for Operator Theory by Douglas: I don't understand the last part of the proof. In order to show that $f$ is continuous, ...
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Are linear and continuous mappings between locally convex vector spaces bounded?

I know that continuity and boundedness of linear mappings between normed vector spaces are equivalent, but does the same hold true for locally convex vector spaces? If so, how can we prove it?
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C*-algebras: Proofs on $C_0(X)$

I'm looking to prove the following but am stuck, please can you help me? $C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact ...
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Regarding integral operators being contractions

I have two half-questions that tie into one another. Suppose $T$ is an operator on $C([0, 1])$ defined by $$(Tu)(t) = \int_0^t (u(x))^2dx.$$ Show that T is not a contraction on the closed unit ball ...
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Open sets in the unitary group $U(\mathcal{H})$ of a Hilbert space $\mathcal{H}$.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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Find Riesz representation of $\phi=f({1\over 2})$

"Let $\rho$ be a space of complex polynomial and define $<f,g>={1\over 2\pi}\int_{0}^{2\pi}f(e^{it})\overline{g(e^{it})}dt$ for $f,g:\rho\to \Bbb{C}$. Let $\phi$ be a linear functional on $\rho$...
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Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...
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Calculate the matrix of a linear opertor that transforms a vector to a Hankel matrix

I would like to calculate the matrix associated to a linear operator $\mathbf{R}$ that transforms a vector $\mathbf{x}\in\mathbb{R}^N$ into a Hankel matrix $\mathbf{H}\in\mathbb{R}^{N-Q+1\times Q}$ ...
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Derivative of linear transformation with confusing moment

After reading this part of Rudin's book i have one question: $A'(\mathbf{x})=A$ seems to me little bit weird because: 1) $A'(x)$ - it's derivative of operator $A$ at point $\mathbf{x}\in \mathbb{R^n}$...
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Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$
Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...