In this work, we study a comparison of norms in non-commutative spaces of τ -measurable operators associated with a semifinite von Neumann algebra. In particular, we obtain Nazarov–Podkorytov type lemma Nazarov et al. (Complex analysis, operators, and related topics. Operatory theory: advances, vol 113, pp 247–267, 2000) and extend the main results in Astashkin et al. (Math Ann, 2023. https://doi.org/10.1007/s00208-023-02606-w ) to non-commutative settings. Moreover, we complete the range of the parameter p for 0

This paper provides a number of examples of relatively weakly compact sets in Orlicz spaces. We show some results arising from these examples. Particularly, we provide a criterion which ensures that some Orlicz function is increasing more rapidly than another (in a sense of T. Ando). In addition, we point out that if a bounded subset K of the Orlicz space LΦ is not bounded by the modular Φ, then it is possible for a set K to remain unbounded under any modular Ψ increasing more rapidly than Φ.

Let f be an arbitrary integrable function on a finite measure space (X,Σ, ν). We characterise the extreme points of the set Ω (f) of all measurable functions on (X,Σ, ν) majorised by f, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.

In this paper, we investigate the conditional expectation on the non-commutative H^(r,s)_p(𝒜;ℓ _∞) and H_p(𝒜;ℓ _1) spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.

Considering the commutative case, we know that a maximal function f = sup(n)vertical bar f(n)vertical bar belongs to L-p(mu) if and only if there is a factorization f(n) = cz(n) = z(n)c for all n is an element of N, where c is an element of L-p(mu) and sup(n)parallel to zn parallel to(infinity) < infinity. The theory of vector-valued noncommutative L-p-spaces are introduced first time by Pisier in 1998. Pisier considered the case M is hyperfinite. This theory solved maximal function's problem in noncommutative case. Later in 2002 Junge and Xu introduced general case. By using these noncommutative vector valued L-p-spaces Junge solved noncommutative version of Doob's maximal inequality problem in general case.The noncommutative vector-valued Hardy spaces were introdused in [2]. In this paper, we consider maximal function's problems on noncommutative Hardy spaces. For this reason we introduce a noncommutative vectorvalued symmetric Hardy space.Our aim is discover their properties. It is presented another useful proof of completeness of this space. We also obtain factorization theorem like Saito's theorem. The work is mostly theoretical. The results can be used to further develop of noncommuative martingale theory, noncommutative ergodic theory, and operator valued Hardy spaces theory.

In this paper, we extended some inequalities which were proved By F. Kittaneh in [9] to the ?-measurable operators.

We consider a self-adjoint differential operator in Hilbert space. Then the domain of the operator is changed by the perturbation of the boundary conditions so that a given neighborhood “is cleared” from the points of the spectrum of the perturbed operator. For the Sturm–Liouville operator on the segment and the Laplace operator on the square such a possibility is attained via integral perturbations of boundary conditions.

In this paper we consider an initial-boundary value problem for the Klein-Gordon-Pock equation. We prove the uniqueness of the solution and find lateral boundary conditions for the Klein-Gordon-Pock equation.

We consider a self-adjoint differential operator in the Hilbert space. The domain of the operator is changed by the perturbation of the boundary conditions so that a given neighborhood "there are no eigenvalues on neighborhood of zero" from the points of the spectrum of the perturbed operator. For the Sturm-Liouville operator on the segment and the Laplace operator on the square such a possibility is achieved through integral perturbations of boundary conditions. These statements are given with full proofs, and with a possible extension.

Let \(M\) be a von Neumann algebra with a normal faithful semifinite trace \(\tau \). Let \(x_1 ,\ldots ,x_n \) be \(n\, \tau \)-measurable positive operators with respect to \(( {M,\tau })\), and let \(z_1 ,\ldots ,z_n \) be \(n\) expansive operators in \(M.\) We prove that for a concave function \(f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) ,\, f(\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} x_{k} z_{k} } ) \) is submajorised by \(\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} f(x_{k} )z_{k} }\) and the reverse submajorisation holds if \(f\) is a positive convex function with \(f( 0)=0.\)

In this paper we introduce the noncommutative \(H^{(r,s)}_{p}({\mathcal A};\ell _{\infty })\) and \(H_{p}({\mathcal A};\ell _{1})\) spaces, and prove the contractivity of the underlying conditional expectation \(\varPhi \) on these spaces. We also give results on duality and complex interpolation.

In this paper we introduce the noncommutative H^(r,s)_p(𝒜;ℓ _∞) and H_p(𝒜;ℓ _1) spaces, and prove the contractivity of the underlying conditional expectation on these spaces. We also give results on duality and complex interpolation.

In this paper we introduce the noncommutative H-p((r,s)) (A; l(infinity)) and spaces, and H-p(A; l(1)) prove the contractivity of the underlying conditional expectation Phi on these spaces. We also give results on duality and complex interpolation.

We prove commutator inequalities associated with polar decompositions of τ -measurable operators.

Let M be a finite von Neumann algebra with a normal faithful finite trace tau. Let x, y be measurable positive operators with respect to (M, tau), and z is an element of M be an expansive operator. We prove f (z*xz) less than or similar to z*f(x)z for any concave function f : [0, infinity) -> [0, infinity), and z*g(x)z less than or similar to g(z*xz) for any convex function g : [0, infinity) -> [0, infinity) with g(0) = 0.

We proved that if (M, τ) is a semi-finite von Neumann algebra, x and y are τ−measurable operators, E is exact interpolation space for the couple (L1(0,∞), L∞(0,∞)) and f is a increasing continuous function on [0,∞). Then (i) in the case f(0) = 0 and g(t) = f( √ t) is operator convex, ‖f(|x|) + f(|y|)‖E(M) ≤ ‖f(|x + y|) + f(|x − y|)‖E(M) ≤ ‖f(2|x|) + f(2|y|)‖E(M). (ii) in the case h(t) = f( √ t) is concave, 1 8‖f(2|x|) + f(2|y|)‖E(M) ≤ ‖f(|x + y|) + f(|x − y|)‖E(M) ≤ 8‖f(|x|) + f(|y|)‖E(M). Mathematics Subject Classification: 46L52, 46E30, 47A30, 47B15

In this paper, we investigate the conditional expectation on the non-commutative H ( r , s ) p ( A ; (cid:2) ∞ ) and H p ( A ; (cid:2) 1 ) spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces