D.J.'s helpful answer shows the most straightforward way to capture any command's (success) output in a variable - by direct assignment.
As for what you tried:
>$output
>, the output redirection operator only supports redirecting to files specified by their name/path (it also supports suppressing output with >$null) - you cannot use it to save output in a variable.
Therefore, the value of $output would be interpreted as the target file name/path; if variable $output was never created, this is tantamount to > $null, i.e., output suppression.
Also note that the files that > creates are invariably plain-text files that reflect the same output formatting that you would see in the console (terminal), and as such they are not suitable for programmatic processing.
By contrast, the common -OutVariable (-ov) parameter you mention in a comment does allow you to capture a command's output objects in a variable, while not interfering with the command's output.
That is, the command's output is still (also) written to the output stream, and if that output isn't consumed (by another command, a variable assignment, or a redirection), it still prints to the console.
E.g., -OutVariable output saves a cmdlet / advanced function's output in variable $output - note the absence of $ in the -OutVariable argument:
PS> Get-Date -OutVariable output; $output
Thursday, June 27, 2019 10:17:07 PM # direct Get-Date output
Thursday, June 27, 2019 10:17:07 PM # output from $output
Therefore, -OutVariable (-ov) is useful:
if you want to see a command's output in the console while also capturing that output in a variable for later analysis.
if you want to capture an intermediate command's output inside a pipeline without interfering with the pipeline.
A slight caveat re -OutVariable (-ov) is that it:
doesn't create regular PowerShell arrays ([object[]]), but instances of [System.Collections.ArrayList].
creates a - single-element - [System.Collections.ArrayList] even if the command outputs only a single object (as Get-Date does, for instance).
These surprising behaviors are discussed in GitHub issue #3154.
However, given PowerShell's flexible collection handling and its member-access enumeration feature, the behaviors may not always be problematic in practice.
-OutVariablealways creates a[System.Collections.ArrayList]instance - even if only one object is output by the command - but it definitely still happens - and is unlikely to change - see this discussion on GitHub.