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Recently, some Young-type inequalities have been promoted. The purpose of this article is to give further refinements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert-Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2 × 2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2 × 2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2 × 2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas. [ABSTRACT FROM AUTHOR]

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ψ α -contractive and ϝ η -contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant. [ABSTRACT FROM AUTHOR]

In our ongoing study, we explore the concepts of I 3 -Cauchy and I 3 -Cauchy for triple sequences in the context of random 2-normed spaces (RTNS). Moreover, we introduce and analyze the notions of I 3 -convergence, I 3 -convergence, I 3 -limit points, and I 3 -cluster points for random 2-normed triple sequences. Significantly, we establish a notable finding that elucidates the connection between I 3 -convergence and I 3 -convergence within the framework of random 2-normed spaces, highlighting their interrelation. Additionally, we provide an illuminating example that demonstrates how I 3 -convergence in a random 2-normed space might not necessarily imply I 3 -convergence. Our observations underscore the importance of condition (AP3) when examining summability using ideals. Furthermore, we thoroughly investigate the relationship between the properties (AP) and (AP3), illustrating through an example how the latter represents a less strict condition compared to the former. [ABSTRACT FROM AUTHOR]

An operator T acting on a Banach space X obeys property (gt) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T −λ is an upper semi-B-Fredholm with index less than or equal to 0. In this paper we study the stability of property (gt) under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T . Moreover, we study the transfer of property (gt) from a bounded linear operator T acting on a Banach space X and a bounded linear operator S acting on a Banach space Y to their tensor product T ⊗S. [ABSTRACT FROM AUTHOR]

In this article, we consider k-quasi-∗-class A operator T ∈ B(H) such that T X = XS for some X ∈ B(K, H) and prove the Fuglede-Putnam type theorem when adjoint of S ∈ B(K) is k-quasi-∗-class A or dominant operators. Among other things, we prove that two quasisimilar k-quasi-∗-class A operators have equal essential spectra. [ABSTRACT FROM AUTHOR]

In this paper, we study Weyl type theorems for f(T), where T is algebraically class p-wA(s, t) operator with 0 < p ≤ 1 and 0 < s, t, s + t ≤ 1 and f is an analytic function defined on an open neighborhood of the spectrum of T. Also we show that if A, B∗ ∈ B(H) are class p-wA(s, t) operators with 0 < p ≤ 1 and 0 < s, t, s+t ≤ 1, then generalized Weyl's theorem, a-Weyl's theorem, property (w), property (gw) and generalized a-Weyl's theorem holds for f(dAB) for every f ∈ H(σ(dAB), where dAB denote the generalized derivation δAB: B(H) → B(H) defined by δAB(X) = AX -XB or the elementary operator ∆AB: B(H) → B(H) defined by ∆AB(X) = AXB - X [ABSTRACT FROM AUTHOR]

For a bounded linear operator T acting on an complex infinite dimensional Hilbert space H, we say that T is an m-quasi-class A(k) operator for k > 0 and m is a positive integer (abbreviation T ∈ ℚ(A(k),m)) if T*m ((T*|T |2kT)1/k+1 - |T|²) Tm ≥ 0. The famous Fuglede-Putnam theorem asserts that: the operator equation AX = XB implies A*X = XB* when A and B are normal operators. In this paper, we prove that if T ∈ ℚ(A(k),m) and S* is an operator of class A(k) for k > 0. Then TX = XS, where X ∈ B(H) is an injective with a dense range which implies XT* = S*X. [ABSTRACT FROM AUTHOR]