scholarly journals The power inequality on normed spaces

1971 ◽  
Vol 17 (3) ◽  
pp. 237-240 ◽  
Author(s):  
Michael J. Crabb
Keyword(s):  
Linear Operator ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Normed Spaces ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Power Inequality ◽  
Unital Banach Algebra

Let X be a complex normed space, with dual space X′. Let T be a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x ∈ X, f ∈ X′, ‖ x ‖ = ‖ f ‖ = f(x) = 1}, and the numerical radius v(T) of T is defined as sup {|z|: z ∈ V(T)}. For a unital Banach algebra A, the numerical range V(a) of a ∈ A is defined as V(Ta), where Ta is the operator on A defined by Tab = ab. It is shown in (2, Chapter 1.2, Lemma 2) that V(a) = {f(a): f ∈ D(1)}, where D(1) = {f ∈ A′: ‖f‖ = f(1) = 1}.

The power inequality on Banach spaces

1971 ◽  
Vol 69 (3) ◽  
pp. 411-415 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Power Inequality ◽  
Image Position

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined asand the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).

scholarly journals Numerical range estimates for the norms of iterated operators

1970 ◽  
Vol 11 (2) ◽  
pp. 85-87 ◽  
Author(s):  
M. J. Crabb
Keyword(s):  
Linear Operator ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Bounded Linear

Let X be a complex normed space, with dual space X′, and T a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x∊X, f∊ X′, ∥x∥ = ∥f∥ = f(x) = 1}. Let ⃒V(T)⃒ denote sup {⃒λ⃒: λ∊ V(T)}. Our purpose is to prove the following theorem.

scholarly journals NORMA OPERATOR PADA RUANG FUNGSI TERINTEGRAL DUNFORD

2019 ◽  
Vol 2 (2) ◽  
pp. 93
Author(s):  
Solikhin Solikhin ◽  
Susilo Hariyanto ◽  
Y.D. Sumanto ◽  
Abdul Aziz
Keyword(s):  
Linear Operator ◽  
Integral Operator ◽  
Distance Function ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Integral Function ◽  
Lebesgue Integral ◽  
Bounded Linear ◽  
Operator Norms

We are discussed operator norms on spce of Dunford integral function. We show that for a function which Dunford integral, operator from dual space into space of Lebesgue integral  is a bounded linear operator. Furthermore, sets of all bounded linear operator is a linear space and it is a normed space by norm certain. Finally, the distance function generated by the norm is metrix space.

scholarly journals Some convexity theorems for matrices

1971 ◽  
Vol 12 (2) ◽  
pp. 110-117 ◽  
Author(s):  
P. A. Fillmore ◽  
J. P. Williams
Keyword(s):  
Linear Operator ◽  
Operator Theory ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Two Dimensional ◽  
Bounded Linear ◽  
Image Position

The numerical range of a bounded linear operator A on a complex Hilbertspace H is the set W(A) = {(Af, f): ║f║ = 1}. Because it is convex andits closure contains the spectrum of A, the numerical range is often a useful toolin operator theory. However, even when H is two-dimensional, the numerical range of an operator can be large relative to its spectrum, so that knowledge of W(A) generally permits only crude information about A. P. R. Halmos [2] has suggested a refinement of the notion of numerical range by introducing the k-numerical rangesfor k = 1, 2, 3, …. It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex.

scholarly journals Polynomials in a hermitian element

1988 ◽  
Vol 30 (2) ◽  
pp. 171-176 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Hermitian Element ◽  
Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
Author(s):  
Fredric M. Pollack
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Subset ◽  
Bounded Linear Operator ◽  
Convex Sets ◽  
Numerical Range ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Empty Convex ◽  
Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

scholarly journals Denseness of operators which attain their numerical radius

1984 ◽  
Vol 36 (1) ◽  
pp. 130-133 ◽  
Author(s):  
I. D. Berg ◽  
Brailey Sims
Keyword(s):  
Linear Operator ◽  
Compact Operator ◽  
Convex Space ◽  
Bounded Linear Operator ◽  
Uniformly Convex ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Uniformly Convex Space ◽  
Small Norm

AbstractWe show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

scholarly journals RUANG BERNORMA LENGKAP ATAS OPERATOR LINEAR TERBATAS PADA RUANG FUNGSI TERINTEGRAL DUNFORD

2020 ◽  
Vol 3 (1) ◽  
pp. 47-55
Author(s):  
Solikhin Solikhin ◽  
YD Sumanto ◽  
Abdul Aziz ◽  
Susilo Hariyanto ◽  
R. Heri Soelistyo Utomo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Integral Function ◽  
Lebesgue Integral ◽  
Bounded Linear ◽  
Operator Norms

Abstract. We are discussed operator norms on space of Dunford integral function. We show that sets of all bounded linear operator from dual space of Banach space into space of Lebesgue integral function is Banach space. Abstrak. Artikel ini membahas norma operator atas operator linear terbatas pada ruang fungsi terintegral Dunford. Himpunan semua operator linear dari ruang dual atas ruang Banach ke ruang fungsi terintegral Lebesgue merupakan ruang bernorma yang lengkap terhadap norma operator yang diberikan.

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

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