# Sergei TreilBrown University · Department of Mathematics

Sergei Treil

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127

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January 2000 - present

## Publications

Publications (127)

The convex body maximal operator is a natural generalization of the Hardy–Littlewood maximal operator. In this paper we are considering its dyadic version in the presence of a matrix weight. To our surprise it turns out that this operator is not bounded. This is in a sharp contrast to a Doob's inequality in this context. At first, we show that the...

The convex body maximal operator is a natural generalisation of the Hardy Littlewood maximal operator. In the presence of a matrix weight it is not bounded.

This paper deals with families of matrix-valued Aleksandrov–Clark measures {μα}α∈U(n), corresponding to purely contractive n×n matrix functions b on the unit disc of the complex plane. We do not make other apriori assumptions on b. In particular, b may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications...

Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side o...

Let $\bfT$ is a certain tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter commutator the following equivalence holds $ \|\bfT b-b \bfT \| \asymp \|b\|_{bmo^d}$. This result is well-known for many types of bi-parameter commutators, see \cite{FS} and \cite{DLWY} for more details.

The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set...

We consider contractive operators $T$ that are trace class perturbations of a unitary operator $U$. We prove that the dimension functions of the absolutely continuous spectrum of $T$, $T^*$ and of $U$ coincide. In particular, if $U$ has a purely singular spectrum then the characteristic function $\theta$ of $T$ is a \emph{two-sided inner} function,...

For an Ap weight w the norm of the Hilbert Transform in Lp(w), 1<p<∞ is estimated by [w]Aps, where [w]Ap is the Ap characteristic of the weight w and s=max(1,1/(p−1)); as simple examples with power weights show, these estimates are sharp.
A natural question to ask, is whether it is possible to improve the exponent s in the above estimate if one re...

The classical Aronszajn-Donoghue theorem states that for a rank one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set...

This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbol{\mu}^\alpha\}_{\alpha\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc of the complex plane. We do not make other apriori assumptions on $b$. In particular, $b$ may be non-inner and/or non-extreme. The...

In this paper we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surpr...

We show that the classical $A_{\infty}$ condition is not sufficient for a lower square function estimate in the non-homogeneous weighted $L^2$ space. We also show that under the martingale $A_2$ condition, an estimate holds true, but the optimal power of the characteristic jumps from $1 / 2$ to $1$ even when considering the classical $A_2$ characte...

For an $A_p$ weight $w$ the norm of the Hilbert Transform in $L^p(w)$, $1<p<\infty$ is estimated by $[w]_{A_p}^{\alpha}$, where $[w]_{A_p}$ is the $A_p$ characteristic of the weight $w$ and $\alpha = \max(1,1/(p-1))$; as simple examples with power weights show, these estimates are sharp. A natural question to ask, is whether it is possible to impro...

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new results in the perturbation theory. First, an analysis of matrix-valued measures and their Cauchy transforms allows us to get a simple proof of the famous Kato--Rosenblum theorem for rank one perturbati...

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $\lambda>0$ \[ |\{ x \in [0,1)^{d} \; :\; g \geq\lambda \}| \leq e^{- \lambda^{2}/(2^{d}\|S(g)\|_{\infty}^{2})}, \] where $S(g)$ denotes the classical dyadic square function in $[0,1)^{d}$. The estimate is sharp when dim...

We start with considering rank one self-adjoint perturbations Aα = A +α( ⋅ , φ)φ with cyclic vector \(\varphi \in \mathcal{H}\) on a separable Hilbert space \(\mathcal{H}\). The spectral representation of the perturbed operator Aα is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights bei...

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\su...

This paper extends the results from arXiv:1702.04569 about sharp $A_2$-$A_\infty$ estimates with matrix weights to the non-homogeneous situation.

We introduce the so called convex body valued sparse operators, which generalize the notion of sparse operators to the case of spaces of vector valued functions. We prove that Calder\'on--Zygmund operators as well as Haar shifts and paraproducts can be dominated by such operators. By estimating sparse operators we obtain weighted estimates with mat...

In this paper, we give necessary and sufficient conditions for weighted $L^2$ estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le C\|f\|_{L^2(\mathbf{W})} \] where $T$ is formally an integral operator with additional structure, $\mathbf{W}, \mathbf{V...

In this paper we prove the weighted martingale Carleson Embedding Theorem
with matrix weights both in the domain and in the target space.

We give a necessary and sufficient condition for the two weight
$L^p$-estimates for paraproducts in non-homogeneous settings, $1<p<\infty$. We
are mainly interested in the case $p\ne 2$, since the case $p=2$ is a
well-known and easy corollary of the Carleson embedding theorem. The necessary
and sufficient condition is given in terms of testing cond...

We start with considering rank one self-adjoint perturbations $A_\alpha =
A+\alpha(\,\cdot\,,\varphi)\varphi$ with cyclic vector $\varphi\in \mathcal{H}$
on a separable Hilbert space $\mathcal H$. The spectral representation of the
perturbed operator $A_\alpha$ is realized by a (unitary) operator of a special
type: the Hilbert transform in the two-...

We investigate the unconditional basis property of martingale differences in
weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference
measure is not doubling).
Specifically, we prove that finiteness of the quantity $[w]_{A_2}=\sup_I \, <
w>_I < w^{-1}>_I$, defined through averages $ <\cdot >_I$ relative to the
reference measu...

The new type of "bumping" of the Muckenhoupt $A_2$ condition on weights is
introduced. It is based on bumping the entropy integral of the weights. In
particular, one gets (assuming mild regularity conditions on the corresponding
Young functions) the bump conjecture, proved earlier by A. Lerner and
independently by Nazarov--Reznikov--Treil--Volberg,...

We give a self-contained proof of the A 2 conjecture, which claims that the norm of any Calderón–Zygmund operator is bounded by the first degree of the A 2 norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condi-tion by the last three authors. Here we replace thi...

This article was written in 1999, and was posted as a preprint in CRM
(Barcelona) preprint series $n^0\, 519$ in 2000. However, recently CRM erased
all preprints dated before 2006 from its site, and this paper became
inacessible. It has certain importance though, as the reader shall see.
Formally this paper is a proof of the (qualitative version of...

We give a history of the Corona Problem in both the one variable and the several variable setting. We also describe connections with functional analysis and operator theory. A number of open problems are described.

We give a necessary and sufficient condition for an n-hypercontraction to be similar to the backward shift operator on a weighted Bergman space. This characterization serves as
a generalization of the description given in the Hardy space setting, where the geometry of the eigenvector bundles of the
operators is used.

For a unitary operator the family of its unitary perturbations by rank one
operators with fixed range is parametrized by a complex parameter $\gamma,
|\gamma|=1$. Namely all such unitary perturbations are $U_\gamma:=U+(\gamma-1)
(., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b,
|\gamma|=1$. For $|\gamma|<1$ operators $U_\gamma...

In this note we show that if the Corona data depends continuously (smoothly)
on a parameter, the solutions of the corresponding Bezout equations can be
chosen to have the same smoothness in the parameter.

We give a necessary and sufficient condition for an n-hypercontraction to be
similar to the backward shift operator in a weighted Bergman space. This
characterization serves as a generalization of the description given in the
Hardy space setting, where the geometry of the eigenvector bundles of the
operators is used.

Carleson measures are ubiquitous in Harmonic Analysis. In the paper of
Fefferman--Kenig--Pipher in 1991 an interesting class of Carleson measures was
introduced for the need of regularity problems of elliptic PDE. These Carleson
measures were associated with $A_\infty$ weights. In discrete setting (we need
exactly discrete setting here) they were s...

We approach the problem of finding the sharp sufficient condition of the
boundedness of all two weight Calderon--Zygmund operators. We solve this
problem in $L^2$ by writing a formula for a Bellman function of the problem.

We give a simple proof of the Sawyer type characterization of the two weigh
estimate for positive dyadic operators (also known as the bilinear embedding
theorem).

We give a simple proof of the so called reproducing kernel thesis for Hankel
operators

We use the Bellman function method to give an elementary proof of a sharp
weighted estimate for the Haar shifts, which is linear in the $A_2$ norm of the
weight and in the complexity of the shift. Together with the representation of
a general Calder\'{o}n--Zygmund operator as a weighted average (over all dyadic
lattices) of Haar shifts, (cf. arXiv:...

A simple shortcut to proving sharp weighted estimates for the Martingale
Transform and for the dyadic shift of order 1 (and so for the Hilbert
transform) is presented. It is a unified proof for these both transforms.
Key words: Calder\on--Zygmund operators, $A_2$ weights, $A_1$ weights,
Carleson embedding theorem, Bellman function, dyadic shifts, n...

In the theory of singular integral operators significant effort is often
required to rigorously define such an operator. This is due to the fact that
the kernels of such operators are not locally integrable on the diagonal, so
the integral formally defining the operator or its bilinear form is not well
defined (the integrand is not in L^1) even for...

Using the combination of three recent papers we give a direct and short proof of $A_2$ conjecture, which claims that the norm of any Calder\'on-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. These three papers are: a) T. Hyt\"onen "The sharp weighted bound for general Calder\'on-Zygmund operators", b) Nazarov-Treil...

In this paper we investigate the relations between (martingale) BMO spaces, paraproducts and commutators in non-homogeneous martingale settings. Some new, and one might add unexpected, results are obtained. Some alternative proof of known results are also presented. Comment: 39 pages, 1 figure This material is based on the work supported by the Nat...

We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calder\'on-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon--Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoint, and the $A_2$ norm of the...

This article was written in 2005 and subsequently lost (at least by the third author). Recently it resurfaced due to one of the colleagues to whom a hard copy has been sent in 2005. We consider here a problem of finding necessary and sufficient conditions for the boundedness of two weight Calder\'on-Zygmund operators. We give such necessary and suf...

We prove a multiparameter version of a classical theorem of Jones and Journe
on weak-star convergence in the Hardy space $H^1$.

We give an example of an operator that satisfies the curvature condition as defined in Kwon and Treil (Publ Mat 53(2):417–438,
2009), but is not similar to the backward shift S* on the Hardy class H
2. We conclude therefore that the contraction assumption in the similarity characterization given in Kwon and Treil (Publ Mat
53(2): 417–438, 2009) is...

We consider rank one perturbations $A_\alpha=A+\alpha(\cdot,\varphi)\varphi$ of a self-adjoint operator $A$ with cyclic vector $\varphi\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_\alpha$ is given by a singular integral operator of special form. Such operators exhibit what we call...

In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess...

We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators. Comment: 18 pages

We prove a self-improvement property regarding quadratic forms on arbitrary vector spaces. We discuss several consequences
of this result, in particular those concerning dimension-free Lp estimates of certain singular integral operators (Riesz transforms).

The paper deals with the problem of ideals of H∞: describe increasing functions φ⩾0 such that for all bounded analytic functions f1,f2,…,fn,τ in the unit disc D the condition|τ(z)|⩽φ((∑|fk(z)|2)1/2)∀z∈D implies that τ belong to the ideal generated by f1,f2,…,fn, i.e. that there exist bounded analytic functions g1,g2,…,gn such that ∑k=1nfkgk=τ.It wa...

If n is a non-negative integer, then denote by ∂-n
H
∞ the space of all complex-valued functions f defined on $$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$$
We prove bounds on the solution in the corona problem for ∂-n
H
∞. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space H
∞...

The main result of the paper is the theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of (generally infinite-dimensional) subspaces (a holomorphic sub-bundle of a trivial bundle). This sufficient condition is also necessary in the case of finite dimension or codimension of the bundle....

In this paper we are proving that Sawyer type condition for boundedness work for the two weight estimates of individual Haar multipliers, as well as for the Haar shift and other "well localized" operators. Comment: 14 pages

In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to a bounded subharmonic function. Using this function we construct a new related Carleson measure that allows f...

We inspect the relationship between the A(p,q) condition for families of norms on vector valued functions and the A(p) condition for scalar weights. In particular, we will show if we are considering a norm-valued function rho(.) such that, uniformly in all nonzero vectors x, rho(.)(X)(p) is an element of A(p) and rho((.))* (X)(q) A(q), then the fol...

Thisvolumecontainsaselectionofpapersinmodernoperatortheoryanditsapp- cations. Most of them are directly related to lectures presented at the Fourteenth International Workshop on Operator Theory and its Applications (IWOTA 2003) held at the University of Cagliari, Italy, in the period of June 24–27, 2003. The workshop, which was attended by 108 math...

In this paper we consider the matrix-valued $H^{p}$ corona problem in the disk and polydisk. The result for the disk is rather well known, and is usually obtained from the classical Carleson Corona Theorem by linear algebra. Our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solu...

This is a continuation of our earlier paper \cite{PT3}. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $\T$ and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly ap...

In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert space H to have a common complement, i.e. a subspace Z having trivial intersection with X and Y and such that H=X+Z=Y+Z.Unlike the finite-dimensional case the condition is significantly more subtle than simple equalities of dimensions and codi...

We give an example of an operator weight W satisfying the op- erator Hunt-Muckenhoupt-Wheeden A2 condition, but for which the Hilbert transform on L2(W) is unbounded. The construction relates weighted bound- edness with the boundedness of vector Hankel operators. We establish a relationship between the norm of a vector Hankel operator and a certain...

The main result of this paper is the following theorem: Given δ, 0 < δ < 1/3 and
\(n \in \mathbb{N},\) there exists an (n + 1) × n inner matrix function
\(F \in H_{(n + 1)\, \times \,n}^\infty \) such that
$$
I \geq F^* (z)F(z) \geq \delta ^2 I,\quad \forall z \in \mathbb{D},
$$ but the norm of any left inverse for F is at least
\(\left[ {\delta /(...

In this paper some new positive results in the Operator Cor-ona Problem are obtained in rather general situation. The main result is that under some additional assumptions about a bounded analytic operator-valued function F in the unit disc D the condition F * (z)F (z) ≥ δ 2 I ∀z ∈ D (δ > 0) implies that F has a bounded analytic left inverse. Typic...

We study in this paper very badly approximable matrix functions on the unit circle
\( \mathbb{T}, \) i.e., matrix functions Φ such that the zero function is a superoptimal approximation of Φ. The purpose of this paper is to obtain a characterization of the continuous very badly approximable functions.
Our characterization is more geometric than alg...

The main result of this paper is that for any unitary (selfadjoint) operatorU with non-trivial absolutely continuous part of the spectrum, there exists a rank-one perturbationK = ba* = (a)b such that the operatorT = U + K satisfies the Linear Resolvent Growth condition (LRG),
||(lI - T) - 1 || \leqslant C/dist(l,s(T)), l Î \mathbbC\s(T),||(\lambda...

The main result of the paper is that there exist functionsf
1,f
2,f inH
∞
satisfying the “corona condition”
$$|f_1 (z)| + |f_2 (z)| \geqslant |f(z)|, z \in \mathbb{D},$$
such thatf
2 does not belong to the idealI generated byf
1,f
2, i.e.,f
2 cannot be represented as f2 ≡ f1g1 + f2g2, g1, g2 ∃ H∞. This gives a negative answer to an old question o...

We prove that given any compact subset of the complex plane containing zero, there exists a Hankel operator having this set as its spectrum.

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderón-Zygmund operator on L<sup>2</sup>(μ)$. We do not assume any kind of doubling condition on the measure $\mu$, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow...

The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators - so called dyadic shifts. We show here that the same is true in any Rn - the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the...

An example is given of an operator weight W that satisfies the dyadic operator Hunt-Muckenhoupt-Wheeden condition A 2d for which there exists a dyadic martingale transform on L 2(W) that is unbounded. The construction relates weighted boundedness to the boundedness of dyadic vector Hankel operators.

It was shown in [1] that if T is a contraction in a Hilbert space with finite defect (i.e., ∥T∥ ≤ 1 and rank (I - T*T) < ∞), and if the spectrum σ(T) does not coincide with the closed unit disk double-struck D sign̄, then the Linear Resolvent Growth condition ∥(λI - T)-1∥≤C/dist(λ, σ(T)), λ ∈ ℂ/σ(T) implies that T is similar to a normal operator. T...

The stochastic optimal control uses the differential equation of Bellman and its solution—the Bellman function. We show how the homonym function in harmonic analysis is (and how it is not) the same stochastic optimal control Bellman function. Then we present several creatures from Bellman's Zoo:a function that proves the inverse Holder inequality,...

We give a sharp estimate of the norm of the S-functions on L
2 (w) in terms of the A
2 “norm” of w.

. We are going to give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform. 0. Introduction Weighted norm inequalities for singular integral operators appear naturally in many areas...

. We are going to give necessary and sufficient conditions for a multivariate stationary stochastic process to be completely regular. We also give the answer to a question of V.V. Peller concerning the spectral measure characterization of such processes. 1. Introduction In this paper we shall give a necessary and sufficient condition for a multivar...

. In the paper we consider Calder'on-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calder'on-Zygmund operator is of weak type if it is bounded in L 2 . We also prove several versions of Cotlar's inequality for maximal singular operator. One version...

We give necessary and sufficient conditions for a multivariate stationary stochastic process to be completely regular. We also give the answer to a question of V.V. Peller concerning the spectral measure characterization of such processes.

In the paper we consider Calder\'{o}n-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calder\'{o}n-Zygmund operator is of weak type if it is bounded in $L^2$. We also prove several versions of Cotlar's inequality for maximal singular operator. One ver...

We give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.

this paper is to consider the boundedness of singular integral operators with Calder'on-Zygmund kernels in L

The main result, the Riesz projectionP+(or, equivalently, Hilbert TransformT), is bounded in the weighted spaceL2(W) whereWis a matrix-valued weight if and only if wheresupremumis taken over all intervalsI. Motivation for this problem comes from stationary processes (Riesz projection is bounded means the angle between “past” and “future” of a stati...

The four block problem is a generalization of Nehari's problem for matrix functions. It plays an important role inH∞-optimal control theory. It is well known that Nehari's problem for a continuous scalar function has a unique solution. However, in the matrix case the situation is quite different. V. V. Peller and N. J. Young (1994,J. Funct. Anal.12...

We are going to show that the classical Carleson embedding theorem fails for Hilbert space valued functions and operator measures.

In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted Lp-spaces are obtained. Invariant A∞ weights are introduced. Several characterizations of invariant A∞ weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators or Hankel operators to be bounded on the...

The main result of the paper is that a system of invariant subspaces of a (completely non-unitary) Hilbert space contraction $T$ with finite defects (rank$(I-T^*T)<\infty$, rank$(I-TT^*)<\infty$) is an unconditional basis (Riesz basis) if and only if it is uniformly minimal. Results of such type are quite well known: for a system of eigenspaces of...

We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function $\left(\begin{array}{cc}\Phi_{11} &\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ on the unit circle the four block problem is to minimize the $L^\infty$ norm of $\left(\begin{array}{cc} \Phi_{11}-F&\Phi_{12}\\\Phi_{21}&\Phi_{...

The goal of this note is to give a very simple proof of the famous Hunt-Muckenhoupt-Wheeden theorem [cf. R. Hung, B. Muckenhoupt and R. Wheeden, Trans. Am. Math. Soc. 176, 227-251 (1973; Zbl 0262.44004)].

It is a well-known fact that for any continuous scalar-valued function φ on the unit circle there is a unique best approximation in the Hardy class H∞ of bounded analytic functions. However, in the matrix-valued case a best approximation by bounded analytic functions is almost never unique. To make it unique, one has to impose additional assumption...

Under consideration was the interaction of the hip joint endoprosthesis pedicle with the femur. Causes of shaking the prosthesis pedicle, fracture of the femur, progressing deforming arthrosis were analyzed. Formulas for calculating the character of interactions of the prosthesis pedicle with the femur are presented, as well as risk factors and rec...