scholarly journals Symplectic Geometry of the Koopman Operator

2021 ◽  
Author(s):  
V. V. Kozlov
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Real Hilbert Space ◽  
Additive Measure ◽  
Symplectic Transformation ◽  
Koopman Operator ◽  
Lebesgue Spectrum ◽  
Quadratic Invariants ◽  
Infinite Set

Abstract We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

A mass-conserved tumor invasion system with quasi-variational degenerate diffusion

2021 ◽  
pp. 1-66
Author(s):  
Akio Ito
Keyword(s):  
Hilbert Space ◽  
Tumor Cells ◽  
Tumor Invasion ◽  
Real Hilbert Space ◽  
Initial Boundary ◽  
Inner Product ◽  
Boundary Problems ◽  
Theory Of Evolution ◽  
Cross Diffusion ◽  
Variational Structure

This paper deals with a nonlinear system (S) composed of three PDEs and one ODE below: [Formula: see text] The system (S) was proposed as one of the mathematical models which describe tumor invasion phenomena with chemotaxis effects. The most important and interesting point is that the diffusion coefficient of tumor cells, denoted by [Formula: see text], is influenced by both nonlocal effect of a chemical attractive substance, denoted by [Formula: see text], and the local one of extracellular matrix, denoted by [Formula: see text]. From this point, the first PDE in (S) contains a nonlinear cross diffusion. Actually, this mathematical setting gives an inner product of a suitable real Hilbert space, which governs the dynamics of the density of tumor cells [Formula: see text], a quasi-variational structure. Hence, the first purpose in this paper is to make it clear what this real Hilbert space is. After this, we show the existence of strong time local solutions to the initial-boundary problems associated with (S) when the space dimension is [Formula: see text] by applying the general theory of evolution inclusions on real Hilbert spaces with quasi-variational structures. Moreover, for the case [Formula: see text] we succeed in constructing a strong time global solution.

scholarly journals Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bin-Chao Deng ◽  
Tong Chen
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Mild Conditions ◽  
Strongly Monotone

LetHbe a real Hilbert space. LetT1,T2:H→Hbek1-,k2-strictly pseudononspreading mappings; letαnandβnbe two real sequences in (0,1). For givenx0∈H, the sequencexnis generated iteratively byxn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn,∀n∈N, whereTwi=1−wiI+wiTiwithi=1,2andB:H→His strongly monotone and Lipschitzian. Under some mild conditions on parametersαnandβn, we prove that the sequencexnconverges strongly to the setFixT1∩FixT2of fixed points of a pair of strictly pseudononspreading mappingsT1andT2.

scholarly journals Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
C. E. Chidume ◽  
C. O. Chidume ◽  
N. Djitté ◽  
M. S. Minjibir
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Iteration Sequence ◽  
Strictly Pseudocontractive Mappings

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

A Finitely Generated Modular Ortholattice

1992 ◽  
Vol 35 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Gunter Bruns ◽  
Michael Roddy
Keyword(s):  
Hilbert Space ◽  
Real Hilbert Space ◽  
Finitely Generated ◽  
Modular Ortholattice

AbstractWe discuss [2] of the same title and offer an alternative example. This example is a subalgebra of the ortholattice of closed subspaces of separable real Hilbert space.

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