The Strong Property (B) for $$L_p$$ Spaces

Given a purely non-atomic, finite measure space $$(\Omega ,\Sigma ,\nu )$$ ( Ω , Σ , ν ) , it is proved that for every closed, infinite-dimensional subspace V of $$L_p(\nu )$$ L p ( ν ) ($$1\le p<\infty $$ 1 ≤ p < ∞ ) there exists a decomposition $$L_p(\nu )=X_1\oplus X_2$$ L p ( ν ) = X 1 ⊕ X 2 , such that both subspaces $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic to $$L_p(\nu )$$ L p ( ν ) and both $$V\cap X_1$$ V ∩ X 1 and $$V\cap X_2$$ V ∩ X 2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on $$L_1$$ L 1 are derived.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 0 1 k s ∫ T Φ ( k x ) d μ , where s and $$\Phi $$ Φ are an outer and Orlicz function respectively and x is a measurable real-valued function over a $$\sigma $$ σ -finite measure space $$(T,\Sigma ,\mu )$$ ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure $$\mu $$ μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Quantitative Uncertainty Principle for Sturm-Liouville Transform

In this paper we consider the Sturm-Liouville transform ℱ(f) on ℝ+. We analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.

Generic non-singular Poisson suspension is of type III1

It is shown that for a dense $G_\delta $ -subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $ -finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1.

Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

Based on the concept of lacunary statistical convergence of sequences of fuzzy numbers, the lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of double sequences of fuzzy-valued functions are defined and investigated in this paper. The relationship among lacunary statistical convergence, uniformly lacunary statistical convergence, equi-lacunary statistical convergence of double sequences of fuzzy-valued functions, and their representations of sequences of α -level cuts are discussed. In addition, we obtain the lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions in a finite measurable space. Finally, the lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions is examined, and it is proved that the inner and outer lacunary statistical convergence in measure are equivalent in a finite measure set for a double sequence of fuzzy-valued measurable functions.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, and Wojciechowski.

Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd

Existence of fixed point of a Frobenius-Perron type operator P : L1 ⟶ L1 generated by a family {φy}y∈Y of nonsingular Markov maps defined on a σ-finite measure space (I, Σ, m) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = {f ∈ L1 : f ≥ 0, and ∥f∥ = 1}, the convergence (in the norm of L1) of the sequence $\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g0. The general result is applied to a family of C1+α-smooth Markov maps in ℝd.