4 problems

Direct-summand conjecture. If

Let $A\in\Gamma_n$, where $\Gamma_n$ denotes the class of accretive $n\times n$ matrices, and let $t$ satisfy $0<t<1$. Fractional-power numerical-radius conjecture. … This conjectu…

Let $A$ be an operator on a Hilbert space of any dimension, and let $w(A)$ denote its numerical radius. The power inequality conjecture. For every positive integer $m$, one has … H…

Let $L_p(\mu)$ be a Lebesgue space, and let $n_1(L_p(\mu))$ denote its rank-one numerical index. Let $\kappa_p$ be the constant defined earlier in the paper. Assume that the dimens…