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Some trace inequalities of Cassels type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

Some reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest and reverses of Hölder and Schwarz trace inequalities are also given.

Cilj ovog rada bio je upoznati se s Hilbert-Schmidtovim operatorima te operatorima s tragom koje nerijetko nazivamo i nuklearnim operatorima. Hilbert-Schmidtovi operatori i operatori s tragom su pod klase kompaktnih operatora i imaju veliku ulogu u raznim područjima, na primjer u kvantnoj fizici, gdje ih štoviše vrlo često možemo susresti. Uvodno smo naveli osnovne definicije linearne algebre, normiranih prostora i operatora na normiranim prostorima, pri čemu smo naglasak stavili na trag operatora, trenutno na konačnodimenzionalnom prostoru. U sljedećem poglavlju smo ukratko ponovili definicije Banachove algebre koju shvaćamo kao analogon Banachovog prostora. U trećem poglavlju smo promatrali kompaktne operatore. S viškom pozornosti, izrekli smo nužne i bitne definicije, jer ipak kompaktni operatori predstavljaju pravi nadskup Hilbert-Schmidtovih operatora i operatora s tragom. Posebnu smo pažnju dali kompaktnim operatorima na Hilbertovim prostorima. Sljedeće poglavlje koje je bilo na redu su Hilbert-Schmidtovi operatori, dakle uz operatore s tragom, glavna tema ovoga rada. Dali smo osnovnu definiciju Hilbert-Schmidtovih operatora i pokazali da navedena definicija ne ovisi o bazi prostora na kojemu se definiraju. Zatim smo potvrdili da je svaki Hilbert-Schmidtov operator kompaktan. Definirali smo i prostor Hilbert-Schmidtovih operatora te izrekli najbitnija svojstva koja ti operatori posjeduju. U zadnjem poglavlju smo pričali o operatorima s tragom, to jest o nuklearnim operatorima. Definirali smo operator s tragom, pri čemu smo pokazali da ne možemo direktno generalizirati trag konačnodimenzionalnog prostora na beskonačnodimenzionalni prostor. Također smo definirali i prostor operatora s tragom, nakon kojeg smo, analogno Hilbert-Schmidtovim operatorima, iznijeli najbitnija svojstva koja operatori s konačnim tragom posjeduju. Zatim smo analizirali dualne prostore, a za sami kraj smo naveli neka poopćenja nuklearnih operatora. The main purpose of this paper was to explain Hilbert-Schmidt operators and trace-class operators which are often called nuclear operators. Hilbert-Schmidt operators and trace-class operators are subclass of compact operators and they can have many applications in different science areas, for example in quantum physics where they are quite common. In the introductory part we had basic definitions of linear algebra, normed spaces and operators on normed spaces where the focus was on trace-class operators, currently in the finite dimensional space. In the next chapter we have repeated the definitions of Banach algebra which is analogue of Banach space. In third chapter we observed compact operators. With extra regard, the necessary and essential definitions were made, since compact operators are the true superset of Hilbert-Schmidt operators and trace-class operators. Special attention was given to compact operators on Hilbert’s space. The next chapter that was observed are Hilbert-Schmidt operators, which were the main topic beside trace-class operators of this thesis. The basic definition of Hilbert-Schmidt operators was given and showed that definition does not depend on the base of the space in which they are defined. Next we showed that every Hilbert-Schmidt operator is compact. We also defined space of Hilbert-Schmidt operators and said the most important properties of those operators. In the final chapter we talked about trace-class operators or nuclear operators. We defined trace-class operators and showed that they cannot be directly generalized by the trace of the finite dimensional space to the infinite dimensional space. Furthermore, space of trace operators were defined where we, analogue to Hilbert-Schmidt operators, said the most important properties of those operators. In the end we analyzed dual vector space’s, and talked about general nuclear operators.

In this paper, we survey some recent trace inequalities for operators in Hilbert spaces that are connected to Schwarz's, Buzano's and Kato's inequalities and the reverses of Schwarz inequality known in the literature as Cassels' inequality and Shisha-Mond's inequality. Applications for some functionals that are naturally associated to some of these inequalities and for functions of operators defined by power series are given. Examples for fundamental functions such as the power, logarithmic, resolvent and exponential functions are provided as well.

The present work deals with some spectral properties of the problem ¶ \medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$, ¶ \vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip ¶ \noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter. ¶ First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space $L^{2}_q[a,b]$. Then, we investigate some asymptotic properties of the spectrum as $\alpha \underset {\lt }{\rightarrow } 1$. We give, in particular, the asymptotic expansion of the first eigenvalue.

We do a preliminary study of the reproducing kernel Hilbert space having as kernel $k^d$, where $d$ is a positive integer and $k$ is the reproducing kernel of the analytic Dirichlet space.

David Hilbert i Erhard Schmidt početkom 20. stoljeća, odnosno 1907. godine, dali su prvu studiju o Hilbert-Schmidtovim operatorima te su po njima ti operatori i dobili ime. Za teoriju Hilbert-Schmidtovih operatora vrlo značajan je John von Neumann, iako se većina njegovih rezultata pripisuje John Wilson Calkinu. U ovom diplomskom radu dan je pregled teorije Hilbert-Schmidtovih operatora koji s nuklearnim operatorima čine istaknutu klasu kompaktnih operatora. Na samom početku dali smo osnovnu definiciju Hilbert-Schmitovih operatora na Hilbertovom prostoru, koju smo kasnije proširili do definicije integralnih Hilbert-Schmidtovih operatora na prostoru pozitivne mjere. Pokazali smo da definicija Hilbert-Schmitovih operatora ovisi samo o prostoru na kojemu se definiraju, ali ne i o bazi tog prostora. Dali smo definiciju Hilbert-Schmidtove norme te pokazali da je skup Hilbert-Schmidtovih operatora zajedno s Hilbert-Schmidtovom normom Banachova algebra te obostrani ideal. Nakon sto smo definirali integralne Hilbert-Schmitove operatore pokazali smo da je svaki Hilbert-Schmitov operator kompaktan operator. Vrlo zanimljiva je teorija o tragu Hilbert-Schmitovih operatora, no prije nego li smo krenuli s tim dijelom teorije morali smo iskazati dvije značajne nejednakosti koje su nam prijeko potrebne za ovaj dio teorije, a to su Hadamardova i Carlemanova nejednakost. Hilbert-Schmidtovi operatori su specifični po svome tragu; za razliku od nuklearnih operatora koji imaju konvergentan trag, trag Hilbert-Schmidtovih operatora definira se za par dvaju operatora. Za funkciju traga Hilbert-Schmidtovih operatora pokazali smo da je to simetrična bilinearna funkcija definirana na produktu skupa Hilbert-Schmidtovih operatora sa samim sobom. Također smo pokazali da je trag dvaju kvazi-nilpotentnih Hilbert-Schmidtovih operatora jednak nuli. Osim traga Hilbert-Schmidtovih operatora, ono što je vrlo zanmljivo u ovom radu je odnos Hilbert-Schmidtovog operatora i analitičke funkcije na okolini njegovog spektra. Pokazali smo da za Hilbert-Schmidtov operator \(A\) i funkciju \(f\) koja je analitička na okolini njegovog spektra vrijedi da je \(f(A)\) Hilbert-Schmidtov operator, a zatim smo pokazali zanimljiv odnos traga ovako definiranih Hilbert-Schmidtovih operatora s jedne strane i vrijednosti analitičkih funkcija u točkama spektra s druge strane. The first study regarding Hilbert-Schmidt operators was given in the early 20th century by David Hilbert and Erhard Schmidt, after whom the operators were also named. This thesis presents a theoretical overview of Hilbert-Schmidt operators, which together with nuclear operators form a distinct class of compact operators. At the beginning of the thesis a basic definition of Hilbert-Schmidt operators on a Hilbert space was given. Afterwards, that definition was extended to define Hilbert-Schmidt integral operators on a positive measure space. Furthermore, the definition of Hilbert-Schmidt operators was shown to depend only on the space it was defined on, and not on the choice of the base of that space. The Hilbert-Schmidt norm was defined, the set of all Hilbert-Schmidt operators was proved to be a Banach algebra, as well as a two-sided ideal. After defining Hilbert-Schmidt integral operators, each Hilbert-Schmidt operator was shown to be compact. The theory regarding the trace of Hilbert-Schmidt operators yielded some very interesting results, however before going into the details of that theory, it was necessary to define and prove Hadamard’s and Carleman’s inequalities. Hilbert-Schmidt operators have a distinct trace; unlike nuclear operators which have a convergent trace, their trace is defined for a pair of operators. The trace function of Hilbert-Schmidt operators was shown to be a symmetric bilinear function defined on the product of the set of Hilbert-Schmidt operators with itself. Furthermore, the trace of two quasi-nilpotent Hilbert-Schmidt operators was proved to be zero. Besides the results regarding the trace of Hilbert-Schmidt operators, an interesting relation between a Hilbert-Schmidt operator and a function which is analytic in a neighborhood of the operators spectrum was also established. Given an operator \(A\), and a function \(f\) , analytic in a neighborhood of the operators spectrum, \(f(A)\) was shown to be a Hilbert-Schmidt operator. Finally, an interesting relation between the trace of said Hilbert-Schmidt operators on one hand and the values of analytic functions in the points of the spectrum was also established.

We consider differential Riccati equations (DREs). These equations arise in many areas and are very important within the field of optimal control. In particular, DREs provide the crucial link between the state and the optimal input in the solution of linear quadratic regulator (LQR) problems. For the approximation of the solutions to DREs we consider the recently introduced splitting methods, with the aim of proving convergence orders in the space of Hilbert--Schmidt operators. The use of this abstract setting yields stronger than usual temporal convergence results, and also implies that these are independent of a subsequent (reasonable) spatial discretization. The main result is that the exponential Lie splitting is first-order convergent, under no artificial regularity assumptions. As side-effects of the analysis, we also acquire concise proofs of the existence and positivity of the exact solutions to abstract DREs, in a more general setting than previously considered.

In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\gamma$-summing or $\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\to F$ such that $w\circ u \circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\to E$ and every bounded operator $w:F\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where one of the spaces $E$ or $F$ is a "Hilbert-Schmidt space", this space seems to have been described only in the easy situation where one of the spaces $E$ or $F$ is a Hilbert space., Comment: 18 pages, in French

We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis.

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form $-d^2/dx^2+V(g;x)$, where the potential is an elliptic function depending on a coupling vector $g\in{\mathbb R}^4$. Alternatively, this operator arises from the $BC_1$ specialization of the $BC_N$ elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on $g$, we associate to this operator a self-adjoint operator $H(g)$ on the Hilbert space ${\mathcal H}=L^2([0,\omega_1],dx)$, where $2\omega_1$ is the real period of $V(g;x)$. For this association and a further analysis of $H(g)$, a certain Hilbert-Schmidt operator ${\mathcal I}(g)$ on ${\mathcal H}$ plays a critical role. In particular, using the intimate relation of $H(g)$ and ${\mathcal I}(g)$, we obtain a remarkable spectral invariance: In terms of a coupling vector $c\in{\mathbb R}^4$ that depends linearly on $g$, the spectrum of $H(g(c))$ is invariant under arbitrary permutations $\sigma(c)$, $\sigma\in S_4$.

The variational representation of the conditional expectation X of a Gaussian signal X given observations Y corrupted by independent white noise is investigated in the general infinite-dimensional setting. Under Hilbert-Schmidt type assumptions it is shown that the filter X can be realized on sample configurations Yω as the extension by continuity of the mapping that gives the solution of a related variational problem.

In recent years much effort are devoted in order to predict the evolution of a discrete time stochastic processes. Few is known on the prediction of a continuous time processes on an entire time interval. For this task we introduce the p-order Banach valued autoregressive processes which appear to be a natural extension of the well-known Rd-valued autoregressive processes to a functional framework. We estimate consistently the parameters of a Hilbertian autoregressive processes when are unknown and we specify the rates. In a wide range of applications this model can be used and is able to handle the prediction successfully.

The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.

Continuous analogs of the strong Szegő limit theorem may be formulated in terms of operators of the form (PTGPT)n−PTGnPT,forn=1,2,…, where G denotes the operator of multiplication by a suitably restricted d×d mvf (matrix-valued function) acting on the space of d×1 vvf’s (vector-valued functions) f that meet the constraint ∫f(μ)∗Δ(μ)f(μ)dμ

Let α,β be ∗-automorphisms of a von Neumann algebra M satisfying the operator equation α+α−1=β+β−1. In this paper we use new techniques (which are useful in noncommutative situations as well) to provide alternate proofs of the results:- If α,β commute then there is a central projection p in M such that α=β on MP and α=β−1 on M(1−P); If M=B(H), the algebra of all bounded operators on a Hilbert space H, then α=β or α=β−1.