We denote $L^{0}$ a set of all (equivalence classes of) extended real valued $m$-measurable functions on $I=[0,\alpha)$, where $0<\alpha\leq\infty$. For any element $x\in L^{0}$ we define $x^{\ast}\left(t\right) =\inf \left\{ \lambda >0:m(|x|>\lambda)\leq t\right\}$, $x^{\ast \ast }(t)=\frac{1}{t}\int_{0}^{t}x^{\ast }(s)ds$ for $t>0$. The \textit{Hardy-Littlewood-P\'olya relation} $\prec$ if given for any $x,y$ in $L^{1}+L^{\infty }$ by
\begin{equation*}
x\prec y\Leftrightarrow x^{\ast \ast }(t)\leq y^{\ast \ast }(t)\text{ for
all }t>0.\text{ }
\end{equation*}
We say that a symmetric space $E$ is \textit{uniformly $K$-monotone} ($E\in(UKM)$)
if for any $(x_n),(y_n)\subset{E}$ such that $x_n\prec{y_n}$ and $\lim_{n\rightarrow\infty}\|x_n\|_{E}=\lim_{n\rightarrow\infty}\|y_n\|_{E}<\infty$ we have $\|{x^*_n-y^*_n}\|_{E}\rightarrow{0}$ as $n\rightarrow\infty$. Analogously, we assume that a symmetric space $E$ is \textit{decreasing uniformly $K$-monotone}, $E\in(DUKM)$ (resp. \textit{increasing uniformly $K$-monotone}, $E\in(IUKM)$) whenever for any $(x_n),(y_n)\subset{E}$ such that $x_{n+1}\prec x_n\prec{y_n}$ (resp. $x_n\prec{y_n}\prec{y_{n+1}}$) for all $n\in\mathbb{N}$ and $\lim_{n\rightarrow\infty}\|x_n\|_{E}=\lim_{n\rightarrow\infty}\|y_n\|_{E}<\infty$ we have $\lim_{n\rightarrow\infty}\|{x^*_n-y^*_n}\|_{E}={0}$.

Recall, an operator $T$ from a Banach function space $(X,\|\cdot\|_X)$ into a Banach function space $(Y,\|\cdot\|_Y)$ is said to be \textit{positive contraction} if its norm is at most one and it satisfies the property $T(x)\geq{0}$ whenever $x\geq{0}$. Moreover, an admissible operator for a Banach couple $(L^1,L^\infty)$ is said to be a \textit{substochastic operator} whenever it is positive contraction on $L^1$ and $L^\infty$ (equivalently $T$ is \textit{substochastic} if and only if $T(x)\prec{x}$ for all $x\in L^1+L^\infty$) see [Bennett and Sharpley, 1988].

We investigate complete criteria under which increasing uniform $K$-monotonicity and lower locally uniform $K$-monotonicity are equivalent in symmetric spaces. Further, we present a connection between $K$-monotonicity properties and the convergence of a sequence of substochastic operators in a norm of the symmetric space. Letting additional assumption in symmetric spaces, we prove compactness of admissible operators. For more details please see [Ciesielski and Lewicki, 2024].


\noindent\textbf{References}

C. Bennett and R. Sharpley, \textit{Interpolation of operators}, Pure and Applied Mathematics Series 129, Academic Press Inc.,1988.

M. Ciesielski and G. Lewicki, \textit{Substochastic operators in symmetric spaces}, (in preparation) (2024).