This paper focuses on establishing new upper bounds for the numerical radius of operators on Hilbert spaces by utilizing the Moore-Penrose inverse and the generalized Cartesian decomposition. The obtained estimates enhance the existing body of knowledge and are systematically compared with results from the current literature. Our findings not only extend but also unify several recent contributions, offering a broader and more cohesive framework for understanding numerical radius inequalities. Through the application of the generalized Cartesian decomposition, we provide deeper insights into the behavior of numerical radii, building upon previous research and opening new directions for further investigation in this field.
Exploring the Sylvester tensor equation not only enhances our understanding of complex systems in various fields but also furnishes us with powerful mathematical instruments to tackle practical challenges. In this study, relying on distinct inequality relationships among the indices that define the positions of elements, we carry out a comprehensive and profound investigation into the solvability of the Sylvester tensor equation by leveraging the theory of block tensors. Furthermore, via rigorous mathematical derivations, we uncover the analytical solution to this arduous equation and offer a thorough analysis of the dimensionality of the solution space.
In this paper, we give necessary and sufficient conditions for weighted translation operators on hypergroups to be topologically multiply recurrent, disjoint topologically transitive and simultaneously transitive. In particular, disjoint topological transitivity and simultaneous transitivity are equivalent in our case.
In this note, we present several refinements of Audenaert's inequality and his generalized results using log-convexity of some functions depending on special cases of unitarily invariant norms, the trace norm and the operator norm. At the same time, we provide a modification to an alternative proof, obtained by Al-Khlyleh and Alrimawi, of the generalized Audenaert's inequality, but for the operator norm.
LetX andY be two infinite dimensional Banach spaces and B(X) (resp.B(Y )) be the algebra of bounded linear operators onX (resp. Y ). For T E B(X ) and x E X , sigma T (x) denotes the local spectrum of T at x. Fix an integer k 2, and let A1 * A2 * ... * Ak stand for a generalized product of any k operators A1, A2, ... , Ak E B(X ).Given two nonzero vectors x0 E X and y0 E Y , in this paper, we characterize all surjective maps phi : B(X)-> B(Y ) which satisfy sigma A1*A2*...*Ak(x0) = sigma phi(A1)*phi(A2)*...*phi(Ak )(y0) for any A1, A2, ... , Ak E B(X ).
Let omega(p)(X) denote the p-operator radius of a bounded linear operator X on a finite dimensional Hilbert space H, where 0 < p 2. In this article, we present p-operator radii generalizations of various numerical radius commutator inequalities, including omega(SX +XS) 2 root 2 omega(S) IIXII, omega(SX* +X*S) 2 omega(S) . IIXII, and the arithmetic-geometric mean inequality: omega(XSY*)<= 1/ 2 omega (|X|S-2+S|Y |(2)), under various conditions on X and Y.
The aim of this paper is to analyze K-frames generated by a bounded linear operator on a separable Hilbert space H. First, we establish some lower bounds for the norm of an operator T when the sequence {T(n)g}(g is an element of G,n >= 0) satisfies the lower K-frame bound for some set G subset of H .Furthermore, we derive a necessary condition for the sequence {T(n)g}(g is an element of G,n >= 0) to be a K-frame. As a consequence, we prove that the hypercyclic operator T with a hypercyclic vector in the range of K cannot generate a K-frame. Additionally, under certain conditions, we construct a Parseval iterative K-frame using an operator. Finally, we determine the form of the K-dual for K-frames generated by an operator.
The Berezin range of a bounded operator A acting on a reproducing kernel Hilbert space H is the set Ber(A) :={Ak(tau), k(tau) : tau is an element of Theta}, where k(tau) is the normalized reproducing kernel for H at tau is an element of Theta. The Berezin radius (number) and the Berezin norms of an operator A are defined by ber (A) := sup(tau is an element of Theta)|tau, k(tau) >divided by, || A ||(ber ,1):= sup(tau ,mu is an element of Theta) |tau, k(mu) >divided by, and || A ||(ber ,2) := sup(tau is an element of Theta)|| Ak(tau)|| respectively. In this paper, we obtain some Berezin radius upper bounds for Hilbert space operators involving the tensor product. Moreover, the obtained upper bounds have been compared with the previously known bounds to demonstrate their reliability.
The order bounded Stevic-Sharma operators between weighted Dirichlet spaces are characterized, which generalizes the previous result obtained by Lin and his colleagues.
It is well known that the Hilbert matrix operator H is bounded from H-infinity to the mean Lipschitz spaces Lambda(p)(1/p) for all 1 < p < infinity. In this paper, we prove that the range of H acting on H-infinity is contained in a certain Zygmund-type space, denoted by Lambda(1.)(1)& lowast;. We also provide explicit upper and lower bounds for the norm of H as an operator from H(infinity)to Lambda(1.)(1)& lowast;. Moreover, we characterize the positive Borel measures mu for which the generalized Hilbert matrix operator H-mu is bounded from H(infinity )to the Hardy space H-q.
In this article, by using the matrix-valued analog of a factorization property of free polynomials, we offer an alternate approach to the structure of matrix-valued hermitian free polynomials that are xy-convex.
We completely characterize the boundedness of the Volterra-type integration operators T-g acting from the weighted Bergman spaces A(omega )(p)to A(nu)(q) for all 0< p,q
Using the factorizations of suitable operators, we establish several identities that give simple and direct understandings as well as provide the remainders and optimizers of the sharp generalized uncertainty principles.
Let N(center dot) be a norm on a unital C*-algebra U. For s and t are both nonnegative reals and s + t > 0, we introduce a family of non-negative real-valued functions on U, defined by v((N,(s,t))) (x) = sup N-theta is an element of R (R-(s,R-t) e(i theta)x , (x E U). Here, R-(s,R-t) (e(i theta)x) = se(i theta)x +t(e((i theta)x)* for all x is an element of U. Some basic properties and other useful characterizations of this family of functions are presented. As a special case of this family of functions, some results involving the weighted algebraic numerical radius are obtained. Additionally, we establish the equivalence between the numerical radius v(x) and the norm v((s,t)) (x).
Let T-n(F-2) be the ring of n x n upper triangular matrices over the Galois field F-2 of two elements. In this paper we characterize strong commutativity preserving additive maps psi: T-n(F-2) -> T-n(F-2) on invertible matrices for n = 2 and n >= 5. This result completes a recent result obtained by Chooi et al. in [14] and yields a comprehensive structural characterization of strong commutativity preserving additive maps on rank k upper triangular matrices over division rings. Some irregular forms are included to exemplify the complexity in structure of strong commutativity preserving additive maps psi : T-n(F-2) -> T-n(F-2) on invertible matrices for n = 3 and 4.
A frame in Rn is a possibly redundant set of vectors {fi}i is an element of I that span Rn . A tight frame in Rn is a generalization of an orthonormal basis. A factor poset P of a frame is the collection of subsets of I, ordered by inclusion, such that J subset of I is in P if and only if { fj}j is an element of J is a tight frame. In [8], the authors studied the conditions for a given poset of index sets to be the factor poset of a frame. They gave a complete characterization of this "inverse factor poset problem" for R2 and a necessary condition for solving this problem in Rn . In this paper we give sufficient conditions on poset P subset of 2I to be a factor poset of a frame and discuss some combinatorial conditions that are necessary for Rn . We also study how to associate tight frames to the vertices of a given graph G such that G becomes the intersection graph of the resulting frame. By establishing a connection between poset characteristics and graph theory, we generate new tight frames. Further we establish the connection between the independence number of a graph and the maximum number of mutually disjoint index sets of prime tight subframes. We also provide an estimation of the size of the factor poset of a frame when the corresponding graph is a complete t-partite graph.
We obtain some bounds for the second smallest and second largest eigenvalues of a Hermitian matrix. Some additional bounds for the second extreme eigenvalues of positive definite matrices are also discussed here.
