Related papers: BMO-estimates for non-commutative vector valued Li…

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the…

Let (M,\mu) be a sigma-finite measure space. Let (T_t) be a semigroup of positive preserving maps on (M,\mu) with standard assumptions. We prove a H_1-BMO duality theory with assumptions only on T_t. The BMO is defined as spaces of…

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms. We prove that if $f$ belongs to the Besov…

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm{VN}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of…

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm{B}(\mathrm{L}^2(X))$ of bounded operators on the…

We describe a Markov dilation for any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$.

We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the…

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a Lipschitz function. If $A,B\in\mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M}),$ then…

We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following…

We start with the Birman--Solomyak approach to define double operator integrals and consider applications in estimating operator differences $f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the Birman--Solomyak approach to the…

For $2\le p<\infty$ we show the lower estimates \[ \|A^{\frac 12}x\|_p \kl c(p)\max\{\pl \|\Gamma(x,x)^{{1/2}}\|_p,\pl \|\Gamma(x^*,x^*)^{{1/2}}\|_p\} \] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps…

We define functions of noncommuting self-adjoint operators with the help of double operator integrals. We are studying the problem to find conditions on a function $f$ on ${\Bbb R}^2$, for which the map $(A,B)\mapsto f(A,B)$ is Lipschitz in…

Ito's construction of Markovian solutions to stochastic equations driven by a L\'evy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the…

Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear…

For a pair $(A,B)$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function $f$ on the Euclidean space ${\Bbb R}^2$ that belongs to the inhomogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$, we…

We consider a class of nonvariational linear operators formed by homogeneous left invariant Hormander's vector fields with respect to a structure of Carnot group. The bounded coefficients of the operators belong to "vanishing logarithmic…

In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$…

We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this…

Calder\'on-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov…

We consider semi-group BMO spaces associated with an arbitrary $\sigma$-finite von Neumann algebra $(\mathcal{M}, \varphi)$. We prove that the associated row and column BMO spaces always admit a predual, extending results from the finite…