We deal with two recent conjectures of R.-C. Li [Linear Algebra Appl. 278 (1998) 317– 326], involving unitarily invariant norms and Hadamard products. In the particular case of the Frobenius norm, the first conjecture is known to be true, whereas the second is still an open problem. In fact, in this paper we show that the Frobenius norm is essentially the only invariant norm which may comply wi...

hold with some constant C independent of f? (Unless otherwise specified, all integrals are taken with respect to the standard Lebesgue measure on R.) Denoting w := u−1, we can reformulate the above question as follows: When is the operator T := M√vT0M√w bounded in L ? (Here Mφ stands for the operator of multiplication by φ.) Such weighted norm inequalities arise naturally in many areas of analy...

We consider the Alugthe transform T̃ = |T |1/2U |T |1/2 of a Hilbert space operator T , where T = U |T | is the polar decomposition of T . We prove that the map T 7→ T̃ is continuous with respect to the norm topology and with respect to the ∗–SOT topology. For T in a tracial von Neumann algebra, we show that the Brown measure is unchanged by the Aluthge transform. We consider the special case whe...

Let $T$ be a bounded linear operator on a Hilbert space $mathscr{H}$. We say that $T$ has the hyponormal property if there exists a function $f$, continuous on an appropriate set so that $f(|T|)geq f(|T^ast|)$. We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function $f$ we introduce the $f$-Aluthge tran...

We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov-Bohm solenoids and a regular part, and prove an Aharonov-Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invarian...

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a ...

We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schrödinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanli...

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘Quantum Mechanics’, then a quantum physicist would not be able to tell a ...