REFINEMENTS OF THE CONTINUOUS TRIANGLE INEQUALITY FOR THE BOCHNER INTEGRAL IN HILBERT SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 521-533
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Numerical Radius ◽  
Operator Inequalities ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some refinements of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for norm and numerical radius operator inequalities are provided. A particular case of interest for complex-valued functions is pointed out as well.

scholarly journals Quadratic reverses of the triangle inequality for Bochner integral in Hilbert spaces

2004 ◽  
Vol 70 (3) ◽  
pp. 451-462
Author(s):  
Sever S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some quadratic reverses of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications of complex-valued functions are provided as well.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

On Integration of Vector-Valued Functions

1958 ◽  
Vol 10 ◽  
pp. 399-412 ◽  
Author(s):  
D. O. Snow
Keyword(s):  
Bochner Integral ◽  
Significant Extent ◽  
Vector Valued Functions ◽  
Integrating Vector ◽  
Vector Valued

Among the variety of integrals which have been devised for integrating vector-valued functions the most widely used is that of Bochner (2), perhaps because of the simplicity of its formulation. Other approaches, including one by Birkhoff (1), have yielded more general integrals yet none of them seems to have supplanted the Bochner integral to a significant extent.

On Measurability for Vector-Valued Functions

1963 ◽  
Vol 15 ◽  
pp. 613-621 ◽  
Author(s):  
D. O. Snow
Keyword(s):  
Banach Space ◽  
Infinite Series ◽  
Unconditional Convergence ◽  
Integration Theory ◽  
Bochner Integral ◽  
Norm Topology ◽  
Vector Valued Functions ◽  
Vector Valued

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

Convexity Properties for Weak Solutions of Some Differential Equations in Hilbert Spaces

1965 ◽  
Vol 17 ◽  
pp. 802-807 ◽  
Author(s):  
S. Zaidman
Keyword(s):  
Hilbert Spaces ◽  
Parabolic Equations ◽  
Linear Operators ◽  
Dense Domain ◽  
Simultaneous Extension ◽  
Convexity Properties ◽  
Vector Valued Functions ◽  
Partial Extension ◽  
Vector Valued ◽  
Closed Linear Operators

In this work we obtain a simultaneous extension of Theorems 1.6 and 1.7 in Agmon and Nirenberg (1), together with a partial extension of the result on backward unicity for parabolic equations by Lions and Malgrange (4).Let H be a Hilbert space. (·) and | | are the notations for the scalar product and the norm in this space. Consider in H a family B(t), 0 ≤ t ≤ T, of closed linear operators with dense domain DB(t) (varying) with t. Let L2(0, T, H) be the space of Bochner square-integrable vector-valued functions with values in H. Our main result is the following

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