arXiv subjects

Tamara Bottazzi

Publications and source records attributed to Tamara Bottazzi.

14 recordsLinked to original sources

arXiv2026

Best approximants relative to a C$^*$-subalgebra, joint numerical range and subdifferentials

We study the minimality of $n\times n$ Hermitian matrices $A$ respect to a $C^*$-subalgebra $\mathcal{B}$ of $M_n(\mathbb{C})$ in the spectral norm, that is \[\|A\|\leq \|A+B\|,\ \text{ for every } B\in \mathcal{B}.\] We generalize the notion of the moment of a subspace and relate it to the joint numerical range and the subdifferentials of the maximum eigenvalue. We extend results previously known for the subalgebra of diagonal operators and describe the subdifferential of the maximum eigenvalue in terms of the moment of the corresponding eigenspace. We also characterize $\mathcal{B}$-minimality via moments and subdifferentials, and provide examples.

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arXiv2025

Semi-inner product and angles in Schatten ideals

In this paper, we investigate the Schatten $p$-class ideals for $p >1$ as semi-inner product spaces in the sense of Giles and Lumer. Within this framework, we explore several geometric and analytic notions such as Birkhoff-James orthogonality, $p$-parallelism, and related properties that naturally arise when these structures are interpreted through the lens of the associated semi-inner product. Furthermore, we introduce a novel notion of angle adapted to this context, which generalizes and unifies existing angle definitions in normed spaces. Our results contribute to a deeper understanding of the geometry of the $p$-Schatten class and offer new perspectives on operator behavior in semi-inner product spaces.

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arXiv2024

Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the relationship between minimal self-adjoint operators and the subdifferential of the maximum eigenvalue, initially for matrices and subsequently for compact operators. In order to do it we obtain new formulas of subdifferentials of maximum eigenvalues of compact operators that become useful in these optimization problems. Additionally, we provide formulas for the minimizing diagonals of rank one self-adjoint operators, a result that might be applied for numerical large-scale eigenvalue optimization. [1] On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl.9 (1988), no 4, 905-918

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arXiv2023

Generalized Buzano Inequality

If $P$ is an orthogonal projection defined on an inner product space $\mathcal{H}$, then the inequality $$ |\langle Px, y\rangle|\leq \frac12 [\|x\|\|y\|+|\langle x, y\rangle|] $$ fulfills for any $x,y \in \mathcal{H}$ (see \cite{Dra16}). In particular, when $P$ is the identity operator, then it recovers the famous Buzano inequality. We obtain generalizations of such classical inequality, which hold for certain families of bounded linear operators defined on $\mathcal{H}$. In addition, several new inequalities involving the norm and numerical radius of an operator are established.

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arXiv2023

Minimal self-adjoint compact operators, moment of a subspace and joint numerical range

We define the (convex) joint numerical range for an infinite family of compact operators in a Hilbert space H. We use this set to determine whether a self-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimal respect to the set of diagonals in a fixed basis E of H in the operator norm, that is ||A|| <= ||A+D||, for all diagonal D. We also describe the moment set m_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms of joint numerical ranges and obtain equivalences between the intersection of moments of two subspaces and of its two related joint numerical ranges. Moreover, we relate the condition of minimality of A or the intersection of the moments of the eigenspaces of ||A|| and -||A|| to the intersection of the joint numerical ranges of two finite families of certain finite hermitian matrices. We also study geometric properties of the set m_S such as extremal curves related with the basis E. All these conditions are directly related with the description of minimal self-adjoint compact operators.

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arXiv2021

Unitary subgroups and orbits of compact self-adjoint operators

Let H be a separable Hilbert space, and D(B(H))^ah the anti-Hermitian bounded diagonals in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators U_kd = {u in U(H): such that u-e^D in K(H) for D in D(B(H))^ah} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob={ e^K b e^{-K} : K in K(H)^ah } of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on Ob defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space K(H)^ah / D(K(H)^ah). Then for every c in Ob and x in T(\ob)_c there exist a minimal lifting Z_0 in B(H)^ah (in the quotient norm, not necessarily compact) such that g(t)=e^{t Z_0} c e^{-t Z_0} is a short curve on Ob in a certain interval.

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arXiv2020

On $A$-parallelism and $A$-Birkhoff-James orthogonality of operators

In this paper, we establish several characterizations of the $A$-parallelism of bounded linear operators with respect to the seminorm induced by a positive operator $A$ acting on a complex Hilbert space. Among other things, we investigate the relationship between $A$-seminorm-parallelism and $A$-Birkhoff-James orthogonality of $A$-bounded operators. In particular, we characterize $A$-bounded operators which satisfy the $A$-Daugavet equation. In addition, we relate the $A$-Birkhoff-James orthogonality of operators and distance formulas and we give an explicit formula of the center mass for $A$-bounded operators. Some other related results are also discussed.

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arXiv2020

Generalized numerical radius and related inequalities

They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving w_N. We also study particular cases with a fixed N(.), for instance the p-Schatten norms. In ["A generalization of the numerical radius". Linear Algebra Appl. 569 (2019)], Abu Omar and Kittaneh defined a new generalization of the numerical radius. That is, given a norm $N(\cdot)$ on $\bh$, the space of bounded linear operators over a Hilbert space H, and A in B(H) w_N(A)=sup_{θ\in \R}N(Re(e^{iθ}A)). They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius and we develop diverse inequalities involving $w_N$. We also study particular cases when N(.) is the p- Schatten norm with p>1.

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arXiv2020

A study of orthogonality of bounded linear operators

We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced by us and also discuss some of the possible applications in this regard. We also study isosceles orthogonality of bounded (positive) linear operators on a Hilbert space and some of the related properties, including that of operators having disjoint support. We further explore the relations between Birkhoff-James orthogonality and isosceles orthogonality in a general Banach space.

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arXiv2019

Geodesic neighborhoods in unitary orbits of self-adjoint operators of K+C

We study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group U_(K+C) of the unitization of the compact operators K(H)+C, or equivalently, the quotient U_(K+C)/ U_Diag(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitian operators of K(H)+C.

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arXiv2016

A Grüss type operator inequality

In [P. Renaud, "A matrix formulation of Grüss inequality", Linear Algebra Appl. 335 (2001), 95--100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we answer positively the Renaud's open problem.

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arXiv2015

Minimal length curves in unitary orbits of a Hermitian compact operator

We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.

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arXiv2014

Inequalities related to Bourin and Heinz means with a complex parameter

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\{z \in {\mathbb C}: Re(z) \in [1/4;3/4]\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by $s_j(A^tB^{1-t}+B^tA^{1-t})\le s_j(A+B)$, answering in the negative a question made by K. Audenaert and F. Kittaneh.

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arXiv2013

Best approximation by diagonal compact operators

We study the existence and characterization properties of compact Hermitian operators C on a separable Hilbert space H such that ||C|| is less or equal than || C + D ||, for all D in D(K(H)). This property is equivalent to || C || = min{||C+D||: D in D(K(H))} = dist (C,D(K(H))), where D(K(H)) denotes the space of compact diagonal operators in a fixed base of H and ||.|| is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.

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