Wyniki wyszukiwania

1

On a property of weak resolvents and its application to a spectral problem

100%

Uetake Y.

Annales Polonici Mathematici

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1997

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tom 66

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nr 1

263-268

EN

We show that the poles of a resolvent coincide with the poles of its weak resolvent up to their orders, for operators on Hilbert space which have some cyclic properties. Using this, we show that a theorem similar to the Mlak theorem holds under milder conditions, if a given operator and its adjoint have cyclic vectors.

2

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

100%

Uetake Y.

Studia Mathematica

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2001

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tom 147

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nr 1

27-35

EN

Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨·,·⟩_{X}$. For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨(I-zA)^{-1}b,c⟩_{X}$. We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

3

The Lax-Phillips infinitesimal generator and the scattering matrix for automorphic functions

100%

Uetake Y.

Annales Polonici Mathematici

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2007

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tom 92

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nr 2

99-122

EN

We study the infinitesimal generator of the Lax-Phillips semigroup of the automorphic scattering system defined on the Poincaré upper half-plane for SL₂(ℤ). We show that its spectrum consists only of the poles of the resolvent of the generator, and coincides with the poles of the scattering matrix, counted with multiplicities. Using this we construct an operator whose eigenvalues, counted with algebraic multiplicities (i.e. dimensions of generalized eigenspaces), are precisely the non-trivial zeros of the Riemann zeta function. We give an operator model on L²(ℝ) of this generator as explicit as possible. We obtain a condition equivalent to the Riemann hypothesis in terms of cyclic vectors for a weak resolvent of the scattering matrix.

4

Standard Models of Abstract Intersection Theory for Operators in Hilbert Space

63%

Banaszak G., Uetake Y.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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tom 63

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nr 2

149-175

EN

For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper (2011) we constructed a GNS (Gelfand-Naimark-Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semisimplicity hold for that operator. (For the definition of semisimplicity of an operator in Hilbert space, see the Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. An operator satisfying this condition can be constructed by using the method of automorphic scattering of Uetake (2009). Combining this with a result from Uetake (2009), we can show that a Dirichlet L-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.

Ograniczanie wyników

2 Annales Polonici Mathematici

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Studia Mathematica

4 Uetake Y.

1 Banaszak G.

1 2015

1 2007

1 2001

1 1997

On a property of weak resolvents and its application to a spectral problem

Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

The Lax-Phillips infinitesimal generator and the scattering matrix for automorphic functions

Standard Models of Abstract Intersection Theory for Operators in Hilbert Space