Kato's conjecture is a mathematical problem

Mathematical problem

A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...

 named after mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Tosio Kato

Tosio Kato

was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis.Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption of the Second World War, he received his doctorate in...

, of the University of California, Berkeley

University of California, Berkeley

The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...

. Kato initially posed the problem in 1953.

Kato asked whether the square root of certain elliptic operator

Elliptic operator

In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is...

s, defined via functional calculus

Functional calculus

In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...

, are analytic

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

.

The problem remained unresolved for nearly a half-century, until it was jointly solved in 2001 by Pascal Auscher, Steve Hofmann

Steve Hofmann

Steve Hofmann is a mathematician who helped solve the famous Kato's conjecture. Said Hofmann, “It's a problem that has interested me since I was a graduate student... It was one of the biggest open problems in my field and everybody thought it was too hard and wouldn't be solved...

, Michael Lacey

Michael Lacey

Michael Thoreau Lacey is an American mathematician. Lacey received his Ph.D. from the University of Illinois at Urbana-Champaign in 1987, under the direction of Walter Philipp. His thesis was in the area of probability in Banach spaces, and solved a problem related to the law of the iterated...

, Alan McIntosh, and Philippe Tchamitchian. The solution was published in 2002 in the Annals of Mathematics journal.