Customizing split rectangle

Question

I want to make split rectangles, but customize the width of each part separately, moreover I would like to hide the lines which split up the distinct rectangles and make the filled part fade from black to gray to white.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{chains,shapes.multipart}
\usetikzlibrary{shapes,calc}
\usetikzlibrary{automata,positioning}

\tikzset{
myshape/.style={
  rectangle split,
  minimum height=1.5cm,
  rectangle split horizontal,
  rectangle split parts=2, 
  draw,
  text width=2cm,
  anchor=center,
  }
}

\begin{document}    

\begin{tikzpicture}
\node[
  myshape,
  rectangle split part fill={white,black}
  ] 
  (shape1) {};
\end{tikzpicture}

\end{document}

The resulting picture looks as follows:

There are however a few problems: I would like the right part of the picture to be wider than the left (but I can only do it the other way around using text width, which is not what I want).

Moreover instead of just using fill, I would like to use shade right color = black, left color = white and when using this, the line between the two parts of the square will become visible (which should then be corrected).