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Hassan Issa

Publications and source records attributed to Hassan Issa.

6 recordsLinked to original sources
arXiv2026

Negative index, matchings, and nonnegative eigenvalues of tridiagonal stochastic matrices

We study negative eigenvalues of $n\times n$ stochastic matrices whose off-diagonal support is constrained by a sparse graph. The main tool is a matching-based inertia principle: if $G$ is bipartite with matching number $\mu(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues counted with their multiplicities) is denoted by $\nu_{-}(S) $, then \[ \nu_{-}(S)\leq \mu(G). \] In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ \nu_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor. $ Consequently, after ordering the eigenvalues of $P$ in the decreasing order, we have $ \lambda_{\lceil n/2\rceil}(P)\geq0, \ \text{and hence} \ \lambda_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Next, we show that this tridiagonal bound is sharp in every dimension in both reducible and irreducible cases. Finally, we explore some possible extension and raise some open questions.

math.PR↗
arXiv2022

On a numerical construction of doubly stochastic matrices with prescribed eigenvalues

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method.

math.NA↗
arXiv2021

On an integral representation of the normalized trace of the $k$-th symmetric tensor power of matrices and some applications

Let $A$ be an $n\times n$ matrix and let $\vee^k A$ be its $k$-th symmetric tensor product. We express the normalized trace of $\vee^k A$ as an integral of the $k$-th powers of the numerical values of $A$ over the unit sphere $\mathbb{S}^{n}$ of $\mathbb{C}^{n}$ with respect to the normalized Euclidean surface measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over $\mathbb{C}^n$. As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in \cite{cuttler} concerning the monotonicity of products of complete symmetric polynomials. In the process, we give a solution to an open problem that was raised by I. Roven\c{t}a and L. E. Temereanca in \cite{roventa}.

math.CO↗
arXiv2018

The construction of $\Psi^\star$-algebra by commutator method containing the Bergman projection on the unit ball

The manuscript is devoted to the construction of $\Psi^\star$- algebras containing the Bergman projection on the unit ball $B_n$ of $\mathbb C^n$. We consider the $C^\star$-algebra $\mathcal{L}(L^2(B_n))$ of bounded operator acting on the Hilbert space of square integrable functions on $B_n$ with respect to the standard probability measure. We search for spectral invariant Fr\'{e}chet subalgebras of $\mathcal{L}(L^2(B_n))$ containing the Bergman projection $P$ which is defined on $L^2(B_n)$ with values in the space of holomorphic functions. We use the commutator method as introduced by Gramsch. This method generalizes the work of Beals for the spectral invariance of pseudodifferential operators. One need not to work on the H\"{o}rmander classes but can describe the algebra of pseudodifferential operators as continuous commutator between scales of Sobolev spaces. To connect the analytical properties to the geometry of the unit ball, we search for linear tangent vector fields $X$ on the the unit sphere $\partial B_n$ so that the commutator $[X,P]$ has a continuous extension to $L^2(B_n)$. In comparison to the work of Bauer in for the case of the Fischer-Fock space it turns out that the results are completely different. Our first contribution states that the set of vector fields which provides the $\Psi^\star$-algebra for the Fock space fails even to provide a continuity with the Bergman projection in the case of the unit ball. Our second contribution, is the construction of vector fields being tangent to the unit sphere which provides a new $\Psi^\star$-algebra containing the Bergman projection of the unit ball. We prove that every linear vector field on the unit sphere commutes with the Bergman projection on the unit ball.

math.FA↗
arXiv2013

On The Doubly Stochastic Realization Of Spectra

An $n$-list $\lambda:=\left(r; \lambda_2, \ldots, \lambda_n\right)$ of complex numbers with $r>0,$ is said to be realizable if $\lambda$ is the spectrum of $n\times n$ nonnegative matrix $A$ and in this case $A$ is said to be a nonnegative realization of $\lambda$. If, in addition, each row and column sum of $A$ equals $r$, then $\lambda$ is said to be doubly stochastically realizable and in such case $A$ is said to be a doubly stochastic realization for $\lambda$. In 1997, Guo proved that if $\left(\lambda_2,\ldots, \lambda_n\right)$ is any list of complex numbers which is closed under complex conjugation then there exists a least real number $\lambda_0$ with $\max\limits_{2\leq j\leq n}|\lambda_j|\leq\lambda_0\leq 2n\max\limits_{2\leq j\leq n}|\lambda_j|$ such that the list of complex numbers $\{ \lambda_1,\lambda_2,...,\lambda_n\}$ is realizable if and only if $\lambda_1\geq \lambda_0$. In 2020, Julio and Soto showed that the upper bound may be reduced to $(n-1)\max\limits_{2\leq j\leq n}|\lambda_j|$ in the case when at least one of the $\lambda_i$ is real. In this paper, we first describe an algorithm for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix $A$ to be cospectral to a doubly stochastic matrix $B$ and in this case $B$ is shown to be the unique closest doubly stochastic matrix to $A$ with respect to the Frobenius norm. Then, our next results slightly improve the upper bound for the nonnegative realization presented by Julio and Soto and in the case when none of the $\lambda_i$ is real, we also give an improvement of Guo's bound. Then, for doubly stochastic realizations, we obtain an upper bound that improves Guo's bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.

math.CO↗