Questions tagged [operator-theory]
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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Trace of Operators [closed]
Let $H_1$ and $H_2$ be two Hilbert Spaces. Let A be an bounded linear operator between $H_1$ and $H_2$ such that $AA^*$ is traceclass, where $A^*$ denotes the adjoint Operator.
Is it true that we have ...
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Contraction and isometry
Let $X$ and $Y$ be two untial commutative Banach algebras. Suppose $X\subset Y$ and $X$ countractively embeds $Y$. Why is it not the case that the embedding is isometric?
From what I see, since $X$ is ...
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We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective.
We are in the space $X = C[1/2, b]$ for some $b < 1$. We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective. The operator $A: X \to X$ is given as $(Af)(x) ...
user1316790
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Proving that operator in $L^2[0,1]$ is compact
I need help with some functional analysis:
Let $A$ be a continuous linear operator on $L^2[0,1]$ and for any $f \in L^2[0,1]$ the function $Af$ is Lipschitz continuous. Show that $A$ is compact.
It is ...
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Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??
I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form):
$\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$
Can anybody explain to me why ...
user1315539
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Rebuilding of bounded linear operator between Banach spaces
It is possible to rebuild a bounded linear operator between Banach spaces knowing the image only through the dual elements?
Maybe we need some other hypothesis but the statement could be something ...
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strong convergence and unitary evolution in Hilbert space
Consider a family of operators $A(t)$ in Hilbert space. It is known that $A(t) \to 0$, $t \to \infty$ in the strong operator topology. Consider the self-adjoint operator $H$ with purely absolutely ...
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Reference for isomorphism theorem for Banach spaces
Let $X,Y$ be Banach spaces, and a linear continuous operator $T:X \rightarrow Y$ then we have that,
denote by $N =\text{ker }T$ and by $\text{Ran }T$ the closed range of T then,
$$
X / N \cong \text{...
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Operator norm between isomorphic spaces with equivalent norms
Let us consider two Banach spaces $X,Y$ which are isomorphic with equivalent norms, i.e. there exists $C_1,C_2>0$ such that,
$$C_1\| x \|_Y \leq \|x\|_X \leq C_2\| x \|_Y.$$
Now suppose we have a ...
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Spectral Theorem applied to Laplace-Beltrami Operator
Consider $(M,g)$ a compact Riemannian manifold, and $L^2(M)$ the Hilbert space of the square-integrable functions with respect to the Riemannian volume form. I want to study the eigenvalue problem for ...
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Checking if the operator $Tf(s)= \int_{s^2}^{\infty }\frac{f(t)}{t}dt$ is continuous.
Let the operator $T:L^2(0,+\infty )\to L^2(1,+\infty )$ be defined by $$Tf(s)= \int_{s^2}^{\infty }\frac{f(t)}{t}dt$$ To check continuity I decided to find the norm $\| T\| =\text{sup}_{\| f\| =1}\| ...
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Forming real symmetric positive semidefinite matrices from complex matrices.
Let $Q \in \mathbb{C}^{n\times n}$ be any matrix. When can we say that the matrix $A=Q^{t}Q$ (where $Q^{t}$ denotes the transpose of a matrix) is a real symmetric positive semidefinite matrix?
Write $...
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strong limit of operators and and weak limit for function
Consider the family of operators in Hilbert space $A(t)$, $t \in [0, \infty)$, $ \| A(t) \| \leq M$, $\forall t \in [0, \infty) $. Let there be convergence to zero in the strong operator topology $A(t)...
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Spectrum of $T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, 4x_{0}, (3x_{-1}), x_{0}, x_{1}, x_{2},\cdots )$
Find spectrum, eigenvalues and eigenvectors of operator $T:\ell^2(\mathbb{Z})\to \ell^2(\mathbb{Z})$, defined by
$$T(\cdots , x_{-2}, x_{-1}, (x_{0}), x_{1}, x_{2},\cdots )=(\cdots ,x_{-2}, x_{-1}, ...
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Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$
Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
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