Olympiad problems

MIPT student olimpiad spring 2023, 4

Is it true that if two linear subspaces $V$ and $W$ of a Hilbert space are closed, then their sum $V+W$ is also closed?

1993 Miklós Schweitzer, 8

Tags: hilbert space , functional analysis

Let H be a complex separable Hilbert space and denote $B(H)$ the algebra of bounded linear operators on H. Find all *-subalgebras C of $B(H)$ for which for all $A \in B(H)$ and $T \in C$ there exists $S \in C$ that $$TA-AT^{\ast} = TS-ST^{\ast}$$ note: *-algebra is also known as involutive algebra.

1999 Miklós Schweitzer, 6

Tags: real analysis , function , hilbert space

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

1993 Miklós Schweitzer, 7

Tags: hilbert space , functional analysis

Let H be a Hilbert space over the field of real numbers $\Bbb R$. Find all $f: H \to \Bbb R$ continuous functions for which $$f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)$$ $$= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)$$ is satisfied for any $x , y , z \in H$.

1997 Miklós Schweitzer, 8

Tags: functional analysis , hilbert space

Let H be an infinite dimensional, separable, complex Hilbert space and denote $\cal B (\cal H)$ the $\cal H$-algebra of its bounded linear operators. Consider the algebras $l_{\infty} ({\Bbb N}, \cal B (\cal H) ) = $ $\{ (a_n) | A_n \in\cal B (\cal H)$ $(n \in {\Bbb N}), \sup_n ||A_n|| <\infty \}$ $C(\beta {\Bbb N}, \cal B (\cal H) )$ = $\{ f: \beta {\Bbb N} \to \cal B (\cal H)|$ f is continuous $\}$ with pointwise operations and supremum norm. Show that these C*-algebras are not isometrically isomorphic. (Here, $\beta {\Bbb N}$ denotes the Stone-Cech compactification of the set of natural numbers.)