10 problems

Let $M$ be a complex symmetric matrix, and suppose that $M$ is unitarily equivalent to a truncated Toeplitz operator (TTO). Let $B$ be an inner function, let $K_B$ be the correspon…

The $C_{\theta}$-real matrix representation conjecture. If $M$ is a complex symmetric matrix and is unitarily equivalent to a TTO, then there exist an inner function $u$ and a…

Let $I$ be an inner function, let $K_I^2$ denote the corresponding model space, and let a bounded symbol of a truncated Toeplitz operator mean a symbol in $L^\infty(\mathbb{T})$ re…

Gu's conjecture. The following are equivalent: $A_B^\theta$ has a reducing subspace such that the restriction has one-dimensional defects; and $\theta(z)=b(z)u(z^N)$ for some inner…

Let $u$ be an inner function on the unit disc, let $K_u=H^2\ominus uH^2$, and let $\mathcal{T}_u$ be the space of truncated Toeplitz operators on $K_u$. For an inner divisor $v$ of…

Let $u$ be an inner function on the unit disc, let $K_u=H^2\ominus uH^2$, and let $\mathcal{T}_u$ denote the space of truncated Toeplitz operators on $K_u$. Operators in…

Characterization conjecture. $A\in\mathcal{T}(\theta)$ if and only if $A_\alpha$ is $C_\alpha$-symmetric for every nonconstant inner function $\alpha$ dividing $\theta$.

One-component characterization conjecture. The equivalent conditions of Theorem 3 are fulfilled if and only if $\theta$ is one-component.

Sarason's quasisymbol conjecture. Every bounded truncated Toeplitz operator $A$ coincides with $A_\mu$ for some $\mu\in\mathcal C_2(\theta)$. Moreover, nonnegative bounded truncate…

Let $\Theta$ be an inner function. A bounded-symbol conjecture asserts that any bounded truncated Toeplitz operator has a bounded symbol if and only if $\Theta$ belongs to $(CLS)$.…