Oct 10, 2018 · I know that sigma(A+B) = sigma(A) + sigma(B). Could you also say sigma (AB)= sigma(A)sigma(B). Also would the same work for the product version using the big Pi symbol.

In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of …

Sigma (/ ˈsɪɡmə / ⓘ SIG-mə; [1] uppercase Σ, lowercase σ, lowercase in word-final position ς; Ancient Greek: σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek …

Exchange $A$ with $B$ to get $\sigma_p(AB)\setminus\{0\} \supset \sigma_p(BA)\setminus\{0\}$. (b) Assume $\dim X=n=\dim Y$. If $ABy=0$ for some $y\ne 0$, then $A$ or $B$ is singular …

If $A \subseteq B$, then any $\sigma$-algebra ${\cal F}$ containing $B$ will also contain $A$. Thus, the intersection of all $\sigma$-algebras containing $A$ must be a subset of the …

Jul 31, 2023 · Do you know how to prove that any eigenvalue of A + B A + B is the sum of an eigenvalue of A A and an eigenvalue of B B? This is a theorem in Rudin's FA. In my copy it is …

May 16, 2023 · Given the spectrum $\sigma([A,B])$ of the commutator $[A,B] := AB-BA$, can we say anything about $\sigma(A)$ and $\sigma(B)$? In particular, I am interested in learning …

Apr 18, 2024 · A way to use the assumption on the spectrum is to write $A$ and $B$ as either the sum $A_1+A_2$ or the product $A_1A_2$ of a self-adjoint matrix $A_1$ and a unitary matrix …

Aug 17, 2021 · Let \(n=ab\) where \(a>0\), \(b>0\) and \(\gcd(a,b)=1\). Then \(\sigma(n)=\sigma(a)\sigma(b)\). Proof. Since \(a\) and \(b\) have only \(1\) as a common …

Let $\mathcal A$ be a $\mathbb{C}$-algebra with a unit (here $\mathcal{A}$ is the algebra of bounded linear operators on your Banach space). Then $\sigma(AB) \smallsetminus \{0\} = …