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Register a new userFuglede-Putnam type commutativity theorems for $ EP $ operators
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Fuglede-Putnam theorem is not true in general for $ EP $ operators on Hilbert spaces. We prove that under some conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get Fuglede-Putnam type theorem for $ EP $ operators -- however proofs are totally different. Finally, interesting results on $ EP $ operators have been proved using sever
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A bounded linear operator $ A$ on a Hilbert space $ mathcal H $ is said to be an $ EP $ (hypo-$ EP $) operator if ranges of $ A $ and $ A^* $ are equal (range of $ A $ is contained in range of $ A^* $) and $ A $ has a closed range. In this paper, we define $EP$ and hypo-$EP$ operators for densely defined closed linear operators on Hilbert spaces and extend results from bounded operator settings to (possibly unbounded) closed operator settings.
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ partial_t u (x,t) = sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) qquad (x,t) in mathbb{R}^N times, ]- infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.
The main objective of this work is to study generalized Browders and Weyls theorems for the multiplication operators $L_A$ and $R_B$ and for the elementary operator $tau_{A,B}=L_AR_B$.
In this article we introduce a new class of Rolewicz-type operators in l_p, $1 le p < infty$. We exhibit a collection F of cardinality continuum of operators of this type which are chaotic and remain so under almost all finite linear combinations, provided that the linear combination has sufficiently large norm. As a corollary to our main result we also obtain that there exists a countable collection of such operators whose all finite linear combinations are chaotic provided that they have sufficiently large norm.
In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(mathbb{R}^{d};X)to L^{q}(mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1leq pleq qleq infty$ and $m:mathbb{R}^dto mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980s, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for $T_m$ without any smoothness properties of $m$. Under smoothness conditions the boundedness results can be extrapolated to other values of $p$ and $q$ as long as $tfrac{1}{p}-tfrac{1}{q}$ remains constant.
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