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Register a new userStrichartz inequality for orthonormal functions associated with special Hermite operator
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In this article, we prove a Strichartz type inequality %associated with Schrodinger equation for a system of orthonormal functions associated with the special Hermite operator $mathcal{L}=-Delta+frac{1}{4}|z|^{2}-i sum_{1}^{n}left(x_{j} frac{partial}{partial y_{j}}-y_{j} frac{partial}{partial x_{j}}right), $ where $Delta$ denotes the Laplacian on $mathbb{C}^{n}$.
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The Strichartz inequality for the system of orthonormal functions for the Hermite operator $H=-Delta+|x|^2$ on $mathbb{R}^n$ has been proved in cite{lee}, using the classical Strichartz estimates for the free Schrodinger propagator for orthonormal systems cite{frank, frank1} and the link between the Schrodinger kernel and the Mehler kernel associated with the Hermite semigroup cite{SjT}. In this article, we give an alternative proof of the above result in connection with the restriction theorem with respect to the Hermite transform with an optimal behavior of the constant in the limit of a large number of functions. As an application, we show the well-posedness results in Schatten spaces for the nonlinear Hermite-Hartree equation.
In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We also show that if t_0 is a point in the domain of a continuous function f, then f is operator monotone if and only if (f(t) - f(t_0))/(t - t_0) is strongly operator convex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of phi circ f when f is operator convex or strongly operator convex.
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the Lowner--Heinz inequality.
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