# Tagged Questions

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### Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks. ...
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### completely bounded maps - convergence

Let $x$ be a completely bounded map between operator spaces $W \subset \mathbf{B}(\mathcal{H})$ and $V \subset \mathbf{B}(\mathcal{K})$, where $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces, and ...
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### Convergence of operator

I would like to know how to solve the following problem (since I didn't manage to solve it on today's exam): Let $A_h:L^1(a,b)\to L^1(a,b)$ be defined: $$A_h f(x)=\frac{1}{h}\int_x^{x+h} g(t) dt,$$ ...
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### Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1$ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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### Proof of Strong Operator Convergence Theorem

Recall the theorem : $T_n \in B(X,Y)$ where $X,\ Y$ are Banachs, is strongly convergent iff (a) $\parallel T_n \parallel$ is bounded (b) $T_nx$ is Cauchy where $x$ is in total subset ...
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### Strong convergence of multiplication operator

I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
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### Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
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### Limit of sequence of growing matrices

Let $$H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),$$ ...
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### The convergence of the adjoint operator

If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
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### In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?

Could anyone help me with this problem? I don't know where to start. Let $\{ e_n \}_{n=1}^\infty$ be an orthonormal basis in a separable Hilbert space $H$. Denote by $P_n$ the orthogonal ...
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### A criterion for convergence in the operator norm

Let $L:H\rightarrow H$ be a continuous linear operator and $R_n:H \rightarrow H$ a sequence of continuous linear operators, where $H$ is a Hilbert space. If the $\sum_{n=1}^{k} R_n$ converge pointwise ...
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### Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that $$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$$ where $|\cdot|$ denotes the standard Euclidean ...
### Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)
Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that ...