# Tag Info

## New answers tagged operator-theory

### Greens Function Partial Derivative Symmetry

Is this true for any greens function in case we have an self-adjoint operator with real coefficients? No. For example, consider the Green's function for the Laplacian $\nabla^2 u = 0$ on $[0,1]$, ...
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### $A \leq B$ implies $\mathcal{R}(A) \subseteq \mathcal{R}(B)$ for self-adjoint, positive operators

Ryszard has already written an answer which provides a great counterexample. Let me also suggest an alternative approach that may be somewhat more illuminating as to what exactly went wrong with the ...
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### $A \leq B$ implies $\mathcal{R}(A) \subseteq \mathcal{R}(B)$ for self-adjoint, positive operators

The conclusion does not hold. First we provide a counterexample for positive operators, and then modify the counterexample in order to cover the case of positive compact operators. Consider the ...
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### A property of compact operators

Recall the standard fact: continuous operators on $H$ with finite dimensional range are dense in $K(H)$. Since such operators are always expressible as the sum of (continuous) operators with 1-...
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### Is the spectral radius continuous around the left-shift operator?

No, $r(\cdot)$ is in fact continuous at $T$. We first consider the right shift. Let $S$ be the right-shift operator, which is an isometry. Let $0 < \epsilon < 1$ and let $Q$ be any operator with ...
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### Question about proof in Lemma 10.9 in "Lectures on von Neumann algebras"

Recall that $s(\omega’_{\xi_1, \xi_0})$ satisfies $R_{s(\omega’_{\xi_1, \xi_0})}\omega’_{\xi_1, \xi_0} = \omega’_{\xi_1, \xi_0}$. That is, $(x’s(\omega’_{\xi_1, \xi_0})\xi_1|\xi_0) = (x’\xi_1|\xi_0)$ ...
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### Left Hilbert algebra: tracial case (Stratila-Zsido: Exercise 10.5)

What you want is done on page 282. $\def\cA{\mathcal A}$ You have that $\varphi_{\cA}(x^*x)<\infty$. This means that $|x|=L_{\xi_0}$ for some $\xi\in\cA''$. If $x=v|x|$ is the polar decomposition, ...
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### Hille-Yosida Generation Theorem

I assume that $A_n= nA R_{n}$ is the Yosida approximation of $A$ and $T_n(t) = e^{tA_n}$ is the uniformly continuous semi-group generated by the bounded operator $A_n$. For $x \in D(A)$ you have ...
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### Sub - $C^*$-algebra $C^* (a)$ of a $C^*$-algebra $A$ generated by an element $a \in A$.

Let's go all the way back to just talking about rings. Given a ring $A$ and an element $a \in A$ we can consider the subring $\mathbb{Z}[a]$ of $A$ generated by $a$. This can be defined in the ...
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### Boundedness of a matrix operator in a norm

As explained in the comments, you are mixing up two different concepts: boundedness of a matrix and uniform boundedness of a family of matrices. To illustrate this via the example \begin{align*} A:(0,\...
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### Proving an Inequality in Hilbert Space: $\int_0^1 \chi_{[t,\infty)}(T) dt \le T$ for $T\ge 0$

According to your calculations $$\int\limits_0^1\chi_{[t,\infty)}(T)\,dt=f_1(T)$$ where $f_1(t)=t\,\chi_{[0,1]}(t)+\chi_{(1,\infty)}(t).$ We have $T=f(T),$ where $f(t)=t.$ As $0\le f_1(t)\le f(t)$ the ...
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### Is the resolvent of a local operator local?

I have stumbled upon another way to see that the resolvent of a local operator is not necessarily local, by using the Neumann series. Informally, we have \begin{align*} (A-z)^{-1} = -\frac{1}{z} \left(...
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### Does there exist a widely-used operator $\boxdot$ such that $(\theta \boxdot \phi)(x) := \theta(x) \circ \phi(x)$?
"Widely used", not that I'm aware of. But this can be expressed in combinatory logic as $\mathsf{SB}$. This works because $\mathsf{S}$ is the combinator $(a,b,c,d \mapsto a (b d)(c d))$, and ...