We show a Kalton-Weis type theorem for the general case of non-commuting operators. More precisely, we consider sums of two possibly non-commuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible, and moreover sectorial. As an application we recover a classical result on the existence, uniqueness and maximal $L^{p}$-regularity for solutions of the abstract linear non-autonomous parabolic problem.
In this paper, first we characterize closedness of range of the finite sum of weighted composition operators between different Lp-spaces. Then we discuss polar decomposition and invertibility of these operators.
Let $X$ be a Banach space, and $M,N$ be two closed subspaces of $X$. We present several necessary and sufficient conditions for the closedness of $M+N$ ($M+N$ is not necessarily direct sum).
We give necessary and sufficient conditions for the sum of n subspaces of a Hilbert space to be closed. We also present various properties of n-tuples of subspaces with closed sum.
We get a criterion for 0 to be in the essential spectrum of a sum of self-adjoint operators whose pairwise products are compact. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of such operators to be closed.
We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than $π$. We show that if one of the operators admits bounded $H^{\infty}$-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for $L^{p}$-maximal …
Let $\mathcal{H}_1,\ldots,\mathcal{H}_n,\mathcal{H}$ be complex Hilbert spaces and $A_k:\mathcal{H}_k\to\mathcal{H}$ be a bounded linear operator with the closed range $Ran(A_k)$, $k=1,\ldots,n$. It is known that if $A_i^*A_j$ is compact for any $i\neq j$, then $\sum_{k=1}^n Ran(A_k)$ is closed. We show that if all products $A_i^*A_j$, $i\neq j$ are "almost" compact, then the subspaces $Ran(A_1),\ldots,Ran(A_n)$ are essentially linearly independent and their sum is closed.
In the present paper we give results on the closedness and the self-adjointness of the sum of two unbounded operators. We present a new approach to these fundamental questions in operator theory. We also prove a new version of the Fuglede theorem where the operators involved are all unbounded. This new Fuglede theorem allows us to prove (under extra conditions) that the sum of two unbounded normal operators remains normal. Also a result on the normality of the unbounded product of two normal operators is obtained as a consequence of this new "Fuglede theorem". Some interesting examples are als…
In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of the…
The primary purpose of this paper is to investigate the question of invertibility of the sum of operators. The setting is bounded and unbounded linear operators. Some interesting examples and consequences are given. As an illustrative point, we characterize invertibility for the class of normal operators. Also, we give a very short proof of the self-adjointness of a normal operator when the latter has a real spectrum.
We investigate the Hirano invertibility of block-operator matrices in Banach algebras, and obtain the Hirano inverse of matrix $\begin{bmatrix} A&B\\ C&D \end{bmatrix}$ under two types of new perturbation conditions. Furthermore, we provide a new operator matrix decomposed into the sum of tripotent and nilpotent elements on Banach spaces.
The integral operator of the form $$\bigl(Nu\bigr)(x)=\sum_{k=1}^\infty e^{i\langleω_k,x\rangle} \int_{\mathbb R^c}n_k(x-y)\,u(y)\,dy$$ acting in $L_p(\mathbb R^c)$, $1\le p\le\infty$, is considered. It is assumed that $ω_k\in\mathbb R^c$, $n_k\in L_1(\mathbb R^c)$, and $$\sum_{k=1}^\infty\lVert n_k\rVert_{L_1}<\infty.$$ We prove that if the operator $\mathbf1+N$ is invertible, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the analogous representation.
A Hilbert space operator $A\in\B$ is left $(X,m)$-invertible by $B\in\B$ (resp., $B\in\B$ is an $(X,m)$-adjoint of $A\in\B$) for some operator $X\in\B$ if $\triangle_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{m-j}XA^{m-j}=0$ (resp., $δ_{B,A}^m(X)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right)B^{(m-j)}XA^j=0$). No Drazin invertible operator $A\in\B$, with Drazin inverse $A_d$, can be left $(I,m)$-invertible (equivalently, $m$-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin inverrtible operat…
The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more gene…
Let $X$ be a Banach space and $T$ be a bounded linear operator acting in $l_p(\mathbb Z^c,X)$, $1\le p\le\infty$. The operator $T$ is called \emph{locally nuclear} if it can be represented in the form \begin{equation*} (Tx)_k=\sum\limits_{m\in\mathbb Z^c} b_{km}x_{k-m},\qquad k\in\mathbb Z^c, \end{equation*} where $b_{km}:\,X\to X$ are nuclear, \begin{equation*} \lVert b_{km}\rVert_{\mathfrak S_1}\leβ_{m},\qquad k,m\in\mathbb Z^c, \end{equation*} $\lVert\cdot\rVert_{\mathfrak S_1}$ is the nuclear norm, $β\in l_{1}(\mathbb Z^c,\mathbb C)$ or $β\in l_{1,g}(\mathbb Z^c,\mathbb C)$, and $g$ is an …
Let $A$ be a densely defined closed operator in a complex Banach space $X.$ Conditions for left invertibility of operators of the form $\sum_{j=1}^\infty a_j (α_j -A)^{-1}$ are given. Several examples are considered.
Suppose that $m,n\in \mathbb{N}$ and that $A:\mathbb{R}^m\to \mathbb{R}^n$ is a linear operator. It is shown here that if $k,r\in \mathbb{N}$ satisfy $k<r\le \mathrm{\bf rank(A)}$ then there exists a subset $σ\subseteq \{1,\ldots,m\}$ with $|σ|=k$ such that the restriction of $A$ to $\mathbb{R}^σ\subseteq \mathbb{R}^m$ is invertible, and moreover the operator norm of the inverse $A^{-1}:A(\mathbb{R}^σ)\to \mathbb{R}^m$ is at most a constant multiple of the quantity $\sqrt{mr/((r-k)\sum_{i=r}^m \mathsf{s}_i(A)^2)}$, where $\mathsf{s}_1(A)\geqslant\ldots\geqslant \mathsf{s}_m(A)$ are the singula…
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = \sum x_j \otimes x_j, rather than on the coordinate one. The John's decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. This is applied to embeddings of l_\infty^k into finite dimensional spaces.
Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the s…
For the large family of ARMA models with variable coefficients (TV-ARMA), either deterministic or stochastic, we provide an explicit and computationally tractable representation based on the general solution of the associated linear difference equation. Analogous representations are established for the fundamental properties of such processes, including the Wold-Cramér decomposition and their covariance structure as well as explicit optimal linear forecasts based on a finite set of past observations. These results are grounded on the principal determinant, that is a banded Hessenbergian repres…
