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Let $\mu$ be some positive measure on $\mathbb{R}$. For technical reasons, I would like to know if the limit $$\lim_{p\rightarrow\infty}\frac {\ln \|f\|_{L^p(\mu)}}{\ln p}$$ exists in $[0,\infty]$ for …

Let $H$ be a separable Hilbert space. Let $A \subseteq B(H)$ be a finitely generated unital algebra. Let $M$ be the strong operator topology closure of $A.$ Let $B_A$ be the closed ball in $A$ and $B_ …

Let $M_n$ denote the $n$ by $n$ matrices. Consider the homomorphisms $$\phi_{n,kn}: M_n \rightarrow M_{kn}$$ which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$ This gives a sensible way t …

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$ Let $\phi: S \rightarrow B_2.$ Given that the blo …

The claim is true. Any difference in norm must be picked up on the span of $(T_s+T_t)^ne_0.$ So we will apply perturbation theory on that subspace. The value of $\langle(T_s+T_t)^ne_0,e_0\rangle$ sho …

Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel Hilber …

I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere. Let $\mathcal{A} \subset M_n(\mathbb{C})$ be an alg …

Let $\mathcal{H}$ be a separable Hilbert space. Let $v$ be a random variable taking values in $\mathcal{H}$ such that $P(v \perp h) < 1$ for all $h \in \mathcal{H}.$ Suppose we sample an infinite sequ …

Another Try We say a $\mathcal{H}$-valued random variable $h$ is a random vector if $P(h \perp g)<1$ for all $g\in \mathcal{H}.$ If $h_1, h_2, \ldots$ is a sequence independent identically distribut …