This paper presents Lipschitz estimates for pairs of contractions. As an application, we establish a modified version of the Krein trace formula for contractions.

   Summary: We establish higher-order trace formulas for contractions, maximal dissipative, and self-adjoint operators under relaxed conditions. This broadens the range of admissible functions and eliminates previous limitations. In this setting, the spectral shift measures are absolutely continuous, and for contractions, admissible functions now include the Besov class—both being new contributions.

Summary: We show that van Suijlekom's technique of imposing conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of unital completely positive maps equipped with the metrizable BW-topology. As a consequence, we deduce that geometric information can be extracted even from partial spectra of the Dirac operator and truncated -algebras.

Summary: Let U(H) denote the set of unitary operators on a complex separable Hilbert space H, and let 1<p<∞. The paper shows that a function f on the unit circle T is n-times continuously Fréchet Schatten p-differentiable at any U ∈ U(H) if and only if f ∈ C^n(T). Consequently, the paper also establishes that f is n-times continuously Gâteaux differentiable along unitary paths. Moreover, formulas for all higher-order derivatives are expressed in terms of multiple operator integrals. 

Summary: We establish estimates for traces of higher-order derivatives of multivariate operator functions, extending previous results to include higher-order spectral shift measures for tuples of commuting contractions under Hilbert-Schmidt perturbations. 

  Summary: This study presents a new proof of Koplienko's trace formula for pairs of contractions, specifically when the initial operator $T_0$ is normal. We use a linear path and employ a finite-dimensional approach, inspired by Voiculescu's method in Krein's trace formula. This extension of the Koplienko trace formula to a class of contraction pairs involves the Schäffer matrix unitary dilation. Additionally, we derive the trace formula for pairs of self-adjoint and maximal dissipative operators through the Cayley transform. Finally, we extend the Koplienko-Neidhardt trace formula for a specific class of contraction pairs via multiplicative paths, using finite-dimensional approximation. 

Summary: This article establishes a connection between Schmidt subspaces of bounded Hankel operators in vector-valued Hardy spaces and nearly $S^*$-invariant subspaces. We demonstrate that these subspaces possess finite defect in general, offering a concise alternative proof for the characterization of Schmidt subspaces in scalar-valued Hardy space. Our findings complement the work of G'{e}rard and Pushnitski, enhancing understanding of Schmidt subspace structures. 

Summary: We derive estimates and representations for Taylor approximation remainders of the spectral action functional $V\mapsto\tau(f(H_0+V))$ on bounded self-adjoint perturbations. Our results extend those in [A.~Skripka: Taylor asymptotics of spectral action functionals. J. Operator Theory, 80  (2018), no. 1, 113--124], accommodating a broader class of functions and stronger estimates when the resolvent of $H_0$ belongs to the noncommutative $L^n$-space. 

  Summary: This article broadens our earlier study, progressing from the Banach setting to the realm of Quasi-Banach spaces.

 Summary: In a previous study, Marcantognini and Morán derived the Koplienko-Neidhardt trace formula for pairs of contractions and maximal dissipative operators using a multiplicative path. This article builds on their work, proving the existence of higher-order spectral shift functions for these pairs through a similar approach.

 Summary: This article introduces a novel set of conjugations in the scalar-valued Hardy space and characterizes complex symmetric Toeplitz operators $T_{\phi}$ with respect to these conjugations in different scenarios. Additionally, it provides a characterization of a complex symmetric block Toeplitz operator $T_{\Phi}$ on the vector-valued Hardy space using specific conjugations from previous works. 

 Summary: This paper investigates the isometric embedding of the Schatten class $S_q$ into the Schatten class $S_p$ ($1\leq p\neq q\leq \infty$). It concludes that, in most instances, the sole feasible scenario is the isometric embedding of $S_2$ into $S_p$. Our work complements the work of Junge, Parcet, Xu, and others in non-commutative $L_p$-space embedding theory. We employ novel elements from perturbation theory, including the Kato-Rellich theorem, multiple operator integrals, and Birkhoff-James orthogonality, along with meticulous case analysis.

Summary: Koplienko established a trace formula for perturbations of self-adjoint operators by Hilbert-Schmidt class operators. Neidhardt later extended this to unitary operators in 1988. This article offers an alternative proof of the Koplienko-Neidhardt trace formula for unitary operators by reducing the problem to a finite-dimensional one, similar to Voiculescu's proof of Krein's trace formula.

Summary: Liang and Partington recently demonstrated that kernels of perturbed Toeplitz operators exhibit near invariance under the backward shift operator in scalar-valued Hardy space. This article extends their result to a vectorial context, revealing kernels of perturbed Toeplitz operator in terms of backward shift-invariant subspaces, using relevant recent theorems. 

Summary: The article investigates nearly invariant subspaces of finite defect for the backward shift operator in the vector-valued Hardy space. It extends a previous result by Chalendar, Gallardo, and Partington and provides a comprehensive description of almost invariant subspaces for both the shift and its adjoint within the same space.