Operator pada Ruang Hilbert

INDONESIA :

Salah satu permasalahan pada analisis fungsional adalah operator linier. Operator linier didefinisikan: Diberikan X dan Y, dua ruang vektor. Operator T:X → Y disebut operator linier jika memenuhi dua kondisi berikut ini:
T(x+y) = Tx + Ty; ∀x,y ∈ X
T(ax) = aTx ; ∀x ∈ X, a ∈ F
dimana F field
Operator linier tidak hanya terdapat pada ruang vektor tetapi berlaku juga pada ruang norm, ruang metrics dan ruang Hilbert. Manfaat dari adanya operator pada ruang Hilbert banyak sekali diantaranya menimbulkan pemikiran baru tentang fisika quantum. Pembahasan tentang operator ruang Hilbert bermacam-macam, tetapi pada penelitian ini pembahasan operator ruang Hilbert pada ruang Hilbert yang kompleks. Operator pada ruang Hilbert yaitu operator Adjoint, Operator Normal, operator self-adjoit dan operator uniter. Pada penelitian ini memperoleh sifat-sifat yang berlaku pada operator adjoint, operator normal, operator self-adjoint dan operator uniter.

ENGLISH :

One of the topics in functional analysis is a linear operator. Linear operator is defined: Given X and Y, are two vector spaces. Operator T:X → Y is called a linear operator if it satisfies the following two conditions:
T(x+y) = Tx + Ty; ∀x,y ∈ X
T(ax) = aTx ; ∀x ∈ X, a ∈ F
Where F is a field.
Linear operators are not only found in vector space, but applies also to the norm space, metrics space and Hilbert space. Benefits of the operator on Hilbert space, is generatins new ideas about physics quantum. Previous studied have been done of operators on Hilbert space, but this study focuses of operators on a complex Hilbert space. Operators on the Hilbert space are adjoint operator, normal operator, self- adjoit operator and unitary operator. This research acquirs the properties that apply to the adjoint operator, normal operator, self-adjoint operator and unitary operator.