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Frank Oertel

Publications and source records attributed to Frank Oertel.

16 recordsLinked to original sources
arXiv2026

Beyond trace-class and Hilbert-Schmidt -- Interaction between operator ideals and von Neumann algebras in quantum physics

Starting from a thorough analysis of the conjugate $\overline{H}$ of a complex Hilbert space $H$, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fr\'{e}chet-Riesz over to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).

quant-ph↗
arXiv2025

Upper bounds for Grothendieck constants, quantum correlation matrices and CCP functions

Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes either the real or the complex field, we concentrate our search on their smallest upper bound. To this end, we establish a basic framework, built on functions which map correlation matrices to correlation matrices entrywise by means of the Hadamard product, such as the Krivine function in the real case or the Haagerup function in the complex case. By making use of multivariate real and complex Gaussian analysis, higher transcendental functions, integration over spheres and combinatorics of the inversion of Maclaurin series, we provide an approach by which we also recover all famous upper bounds of Grothendieck himself ($K_G^\mathbb{R} \leq \sinh(π/2) \approx 2.301$), Krivine ($K_G^\mathbb{R} \leq \fracπ{2 \ln(1 + \sqrt{2})} \approx 1,782$) and Haagerup ($K_G^\mathbb{C} \leq 1.405$, numerically approximated); each of them as a special case. In doing so, we aim to unify the real and complex case as much as possible and apply our results to several concrete examples, including the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian copula - with foundations of quantum theory and quantum information theory in mind. Moreover, we offer a shortening and a simplification of the proof of the strongest estimation until now; namely that $K_G^\mathbb{R} < \fracπ{2 \ln(1 + \sqrt{2})}$. We summarise our key results in form of an algorithmic scheme and shed light on related open problems and topics for future research.

math.FA↗
arXiv2023

Beyond trace class -- Tensor products of Hilbert spaces and operator ideals in quantum physics

Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fr\'{e}chet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely $p$-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler ($p=2$) and more recently in the framework of general probabilistic spaces ($p=1$), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces $H\otimes_2 (K \otimes_2 L)$ and $(H \otimes_2 K) \otimes_2 L$ (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces $H \otimes_2 K$ (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).

math.FA↗
arXiv2020

Grothendieck's inequality and completely correlation preserving functions -- a summary of recent results and an indication of related research problems

As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach, primarily built on functions which map correlation matrices entrywise to correlation matrices by means of the Schur product, multivariate Gaussian analysis, copulas and inversion of suitable Taylor series. We summarise first results and point towards related open problems and topics for future research.

math.FA↗
arXiv2015

On Jump Measures of Optional Processes with Regulated Trajectories

Starting from an iterative and hence numerically easily implementable representation of the thin set of jumps of a càdlàg adapted stochastic process $X$ (including a few applications to the integration with respect to the jump measure of $X$), we develop similar representation techniques to describe the set of jumps of optional processes with regulated trajectories and introduce their induced jump measures with a view towards the framework of enlarged filtration in financial mathematics.

math.PR↗
arXiv2015

An analysis of the Rüschendorf transform - with a view towards Sklar's Theorem

In many applications including financial risk measurement, copulas have shown to be a powerful building block to reflect multivariate dependence between several random variables including the mapping of tail dependencies. A famous key result in this field is Sklar's Theorem. Meanwhile, there exist several approaches to prove Sklar's Theorem in its full generality. An elegant probabilistic proof was provided by L. Rüschendorf. To this end he implemented a certain "distributional transform" which naturally transforms an arbitrary distribution function $F$ to a flexible parameter-dependent function which exhibits exactly the same jump size as $F$. By using some real analysis and measure theory only (without involving the use of a given probability measure) we expand into the underlying rich structure of the distributional transform. Based on derived results from this analysis (such as Proposition 2.5 and Theorem 2.12) including a strong and frequent use of the right quantile function, we revisit Rüschendorf's proof of Sklar's theorem and provide some supplementing observations including a further characterisation of distribution functions (Remark 2.3) and a strict mathematical description of their "flat pieces" (Corollary 2.8 and Remark 2.9).

math.PR↗
arXiv2012

Restructuring Counterparty Credit Risk

We introduce an innovative theoretical framework to model derivative transactions between defaultable entities based on the principle of arbitrage freedom. Our framework extends the traditional formulations based on Credit and Debit Valuation Adjustments (CVA and DVA). Depending on how the default contingency is accounted for, we list a total of ten different structuring styles. These include bipartite structures between a bank and a counterparty, tri-partite structures with one margin lender in addition, quadri-partite structures with two margin lenders and, most importantly, configurations where all derivative transactions are cleared through a Central Counterparty (CCP). We compare the various structuring styles under a number of criteria including consistency from an accounting standpoint, counterparty risk hedgeability, numerical complexity, transaction portability upon default, induced behaviour and macro-economic impact of the implied wealth allocation.

q-fin.RM↗
arXiv2007

Geometry of polar wedges and super-replication prices in incomplete financial markets

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. By introducing a general convex-analytic framework which includes the class of umbrella wedges in certain Riesz spaces and faces of convex sets (consisting of probability measures), together with a duality theory for polar wedges, we provide a representation of the super-replication price of an unbounded (but sufficiently integrable) contingent claim that can be dominated approximately by a zero-financed terminal wealth as the the supremum of its discounted expectation under pricing measures which appear as faces of a given set of separating probability measures. Central to our investigation is the representation of a wedge $C_Φ$ of utility-based super-replicable contingent claims as the polar wedge of the set of finite entropy separating measures. Our general approach shows, that those terminal wealths need {\it not} necessarily stem from {\it admissible} trading strategies only. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the wedge $C_Φ$: the utility-based restrictions which this wedge imposes on terminal wealth arise only from the investor's preferences to asymptotically large negative wealth.

math.PR↗
arXiv2006

On random measures, unordered sums and discontinuities of the first kind

By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} Φ\circ | ΔX_s |$, where $X : Ω\times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $Φ: \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. Proposition \ref{prop:sum of jumps on R+ with invertible function}). Moreover, our approach enables a natural extension of the jump measure of càdlàg and adapted processes to an integer-valued random measure of optional processes with regulated trajectories which need not necessarily to be right- or left-continuous (cf. Theorem \ref{thm:optional random measures}). In doing so, we provide a detailed and constructive proof of the fact that the set of all discontinuities of the first kind of a given real-valued function on $\R$ is at most countable (cf. Lemma \ref{lemma:right limits and left limits}, Theorem \ref{thm:at most countably many jumps on compact intervals} and Theorem \ref{thm:at most countably many jumps on R+}). By using the powerful analysis of unordered sums, we hope that our contributions fill an existing gap in the literature, since neither a detailed proof of (the frequently used) Theorem \ref{thm:at most countably many jumps on compact intervals} nor a precise definition of sums of jumps seems to be available yet.

math.PR↗
arXiv2006

On utility-based super-replication prices of contingent claims with unbounded payoffs

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at $-\infty$ we prove that the utility-based super-replication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite {\it loss-entropy}. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is the representation of a cone $C_U$ of utility-based super-replicable contingent claims as the polar cone to the set of finite loss-entropy pricing measures. The cone $C_U$ is defined as the closure, under a relevant weak topology, of the cone of all (sufficiently integrable) contingent claims that can be dominated by a zero-financed terminal wealth. We investigate also the natural dual of this result and show that the polar cone to $C_U$ is generated by those separating measures with finite loss-entropy. The full two-sided polarity we achieve between measures and contingent claims yields an economic justification for the use of the cone $C_U$, and an open question.

math.PR↗
arXiv2001

On normed products of operator ideals which contain $\frak{L}_2$ as a factor

We investigate quasi-Banach operator ideal products $({\frak{A}}\circ{\frak{B}},\mathbf{A\circ B})$ which contain $(\frak{L}_2, \mathbf{L}_2)$ as a factor. In particular, we ask for conditions which guarantee that $\mathbf{A\circ B}$ is even a norm if each factor of the product is a 1-Banach ideal. In doing so, we reveal the strong influence of the existence of such a norm in relation to the accessibility of the product ideal and the structure of its factors.

math.FA↗
arXiv2001

The principle of local reflexivity for operator ideals and its implications

We present a survey of past research activities and current results in constructing a mathematical framework describing the principle of local reflexivity for operator ideals and reveal further applications involving operator ideal products consisting of operators which factor through a Hilbert space.

math.FA↗
arXiv1996

Local properties of accessible injective operator ideals

In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which {\it{are accessible}}. The first step is implied by the observation that a "good behaviour" of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of {\it{absolutely summing operators}} (prop. 4.1). By the famous {\it{Grothendieck inequality}}, every operator from $L_1$ to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization - via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence,we obtain a class of injective ideals, which are "quasi-accessible", and with the help of {\it{tensor stability}}, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.

math.FA↗
arXiv1996

Composition of operator ideals and their regular hulls

Given two quasi-Banach ideals \oid{A}{}{} and \oid{B}{}{} we investigate the regular hull of their composition - $(\oid{A}{}{} \circ \oid{B}{}{})^{reg}$. In concrete situations this regular hull appears more often than the composition itself. As a first example we obtain a description for the regular hull of the nuclear operators which is a "reflected" Grothendieck representation:\\ $\oid{N}{}{reg} \stackrel{1}{=} \oid{I}{}{} \circ \oid{W}{}{}$ (theorem 2.1). Further we recognize that the class of such ideals leads to interesting relations concerning the question of the accessibility of (injective) operator ideals.

math.FA↗