<p>In this paper, some generalized Cauchy–Schwarz inequalities for positive sesquilinear maps with values in noncommutative <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-spaces for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are obtained. Bound estimates for their real and imaginary parts are also provided and, as an application, a generalization of the uncertainty relation in the context of noncommutative <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-spaces is given. Next, a new norm on a noncommutative <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-space which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space is proposed and a Cauchy–Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a noncommutative <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-space and into certain operator spaces in several different situations. Some concrete examples are also given.</p>

Cauchy–Schwarz Inequalities for Maps in Noncommutative $L^p$-Spaces

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2026年5月15日

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Giorgia Bellomonte,
Stefan Ivković,
Camillo Trapani

摘要

In this paper, some generalized Cauchy–Schwarz inequalities for positive sesquilinear maps with values in noncommutative $L^p$ $L^{p}$ -spaces for $p>1$ $p > 1$ are obtained. Bound estimates for their real and imaginary parts are also provided and, as an application, a generalization of the uncertainty relation in the context of noncommutative $L^2$ $L^{2}$ -spaces is given. Next, a new norm on a noncommutative $L^2$ $L^{2}$ -space which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space is proposed and a Cauchy–Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative $L^2$ $L^{2}$ -space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a noncommutative $L^p$ $L^{p}$ -space and into certain operator spaces in several different situations. Some concrete examples are also given.

Cauchy–Schwarz Inequalities for Maps in Noncommutative \(L^p\)-Spaces

摘要